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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Exponential and Logarithmic Functions
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Introduction
1. It is now time to view exponentials and logarithms as functionsof a complex variable.
2. For the exponential function, there will be no surprises.3. The logarithm (and with it complex roots) turn out to be more
subtle.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Introduction1. It is now time to view exponentials and logarithms as functions
of a complex variable.
2. For the exponential function, there will be no surprises.3. The logarithm (and with it complex roots) turn out to be more
subtle.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Introduction1. It is now time to view exponentials and logarithms as functions
of a complex variable.2. For the exponential function, there will be no surprises.
3. The logarithm (and with it complex roots) turn out to be moresubtle.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Introduction1. It is now time to view exponentials and logarithms as functions
of a complex variable.2. For the exponential function, there will be no surprises.3. The logarithm (and with it complex roots) turn out to be more
subtle.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition.
For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem.
The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof.
With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y)
=∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y)
=−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez
=∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y)
= ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y)
= ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = x+ iy ∈ C define ez := ex(
cos(y)+ isin(y))
andcall it the exponential function.
Theorem. The exponential function is entire withddz
ez = ez.
Proof. With ez = ex cos(y)+ iex sin(y) we have
∂u∂x
= ex cos(y) =∂v∂y
∂u∂y
= −ex sin(y) =−∂v∂x
Therefore, by the Cauchy-Riemann equations ez is differentiable atevery z ∈ C and
ddz
ez =∂
∂xex cos(y)+ i
∂
∂xex sin(y) = ex cos(y)+ iex sin(y) = ez.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2
= ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)
= ex1ex2[
cos(y1)cos(y2)− sin(y1)sin(y2)++icos(y1)sin(y2)+ isin(y1)cos(y2)
]= ex1ex2
[cos(y1 + y2)+ isin(y1 + y2)
]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]
= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2)
= ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2
= ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez
= ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))
6= 0ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi
= eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi
= ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Some Properties
ez1ez2 = ex1(
cos(y1)+ isin(y1))ex2
(cos(y2)+ isin(y2)
)= ex1ex2
[cos(y1)cos(y2)− sin(y1)sin(y2)+
+icos(y1)sin(y2)+ isin(y1)cos(y2)]
= ex1ex2[
cos(y1 + y2)+ isin(y1 + y2)]= ex1+x2ei(y1+y2)
= ex1+x2+i(y1+y2) = ex1+iy1+x2+iy2 = ez1+z2
ez = ex(cos(y)+ isin(y))6= 0
ez+2πi = eze2πi = ez
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition.
For every nonzero w ∈ C there is a z ∈ C so that ez = w.
Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.
Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.
Proof.
From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.
Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ
= eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.
Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ
= eln(r)+iθ . So z := ln(r)+ iθ is asdesired.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.
Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ .
So z := ln(r)+ iθ is asdesired.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.
Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For every nonzero w ∈ C there is a z ∈ C so that ez = w.
Proof. From the exponential form for nonzero complex numbers, weknow that w = reiθ = eln(r)eiθ = eln(r)+iθ . So z := ln(r)+ iθ is asdesired.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition.
For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z)
= eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn)
= eln(r)eiθ ei2πn = reiθ = z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn
= reiθ = z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ
= z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z
log(ez)
= log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z
log(ez) = log(exeiy)
= ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn)
= x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0, we define the (multiple valued)logarithmic function as
log(z) := ln(r)+ i(θ +2πn)
elog(z) = eln(r)+i(θ+2πn) = eln(r)eiθ ei2πn = reiθ = z
log(ez) = log(exeiy) = ln(ex)+ i(y+2πn) = x+ iy+ i2πn
= z+ i2πn
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition.
For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as
Log(z) := ln(r)+ iθ
log(−1) = log(e0+iπ
)= 0+ i(π +2πn) = (2n+1)πi
Log(−1) = Log(e0+iπ)
= πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as
Log(z) := ln(r)+ iθ
log(−1) = log(e0+iπ
)= 0+ i(π +2πn) = (2n+1)πi
Log(−1) = Log(e0+iπ)
= πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as
Log(z) := ln(r)+ iθ
log(−1) = log(e0+iπ
)= 0+ i(π +2πn) = (2n+1)πi
Log(−1) = Log(e0+iπ)
= πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as
Log(z) := ln(r)+ iθ
log(−1)
= log(e0+iπ
)= 0+ i(π +2πn) = (2n+1)πi
Log(−1) = Log(e0+iπ)
= πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as
Log(z) := ln(r)+ iθ
log(−1) = log(e0+iπ
)
= 0+ i(π +2πn) = (2n+1)πiLog(−1) = Log
(e0+iπ)
= πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as
Log(z) := ln(r)+ iθ
log(−1) = log(e0+iπ
)= 0+ i(π +2πn)
= (2n+1)πiLog(−1) = Log
(e0+iπ)
= πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as
Log(z) := ln(r)+ iθ
log(−1) = log(e0+iπ
)= 0+ i(π +2πn) = (2n+1)πi
Log(−1) = Log(e0+iπ)
= πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as
Log(z) := ln(r)+ iθ
log(−1) = log(e0+iπ
)= 0+ i(π +2πn) = (2n+1)πi
Log(−1)
= Log(e0+iπ)
= πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as
Log(z) := ln(r)+ iθ
log(−1) = log(e0+iπ
)= 0+ i(π +2πn) = (2n+1)πi
Log(−1) = Log(e0+iπ)
= πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z = reiθ 6= 0 with −π < θ ≤ π , we define theprincipal value of the logarithm as
Log(z) := ln(r)+ iθ
log(−1) = log(e0+iπ
)= 0+ i(π +2πn) = (2n+1)πi
Log(−1) = Log(e0+iπ)
= πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem.
On any set of complex numbers z = reiθ withα < θ < α +2π and r > 0, the function log(z) = ln(r)+ iθ is
analytic with derivativeddz
log(z) =1z
.
Proof. Done as an example when we looked at the Cauchy-Riemannequations in polar coordinates.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. On any set of complex numbers z = reiθ withα < θ < α +2π and r > 0, the function log(z) = ln(r)+ iθ is
analytic with derivativeddz
log(z) =1z
.
Proof. Done as an example when we looked at the Cauchy-Riemannequations in polar coordinates.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. On any set of complex numbers z = reiθ withα < θ < α +2π and r > 0, the function log(z) = ln(r)+ iθ is
analytic with derivativeddz
log(z) =1z
.
Proof.
Done as an example when we looked at the Cauchy-Riemannequations in polar coordinates.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. On any set of complex numbers z = reiθ withα < θ < α +2π and r > 0, the function log(z) = ln(r)+ iθ is
analytic with derivativeddz
log(z) =1z
.
Proof. Done as an example when we looked at the Cauchy-Riemannequations in polar coordinates.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. On any set of complex numbers z = reiθ withα < θ < α +2π and r > 0, the function log(z) = ln(r)+ iθ is
analytic with derivativeddz
log(z) =1z
.
Proof. Done as an example when we looked at the Cauchy-Riemannequations in polar coordinates.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition.
A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).
Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.
Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.
Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).
Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.
Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.
Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).
Example.
The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.
Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.
Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).
Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.
Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.
Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).
Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.
Definition.
A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.
Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).
Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.
Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function.
Abranch point is a point that is on all branch cuts for a particularfunction.
Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).
Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.
Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.
Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).
Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.
Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.
Example.
Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).
Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.
Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.
Example. Any ray θ = α is a branch cut for the logarithm function.
The point z = 0 is a branch point for the logarithm.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. A branch of a multiple valued function f is any singlevalued function F that is analytic in some domain and at each point zin the domain the value F(z) is one of the values of f (z).
Example. The function Log(z) := ln(r)+ iθ on r > 0, −π < θ < π iscalled the principal branch of the logarithm.
Definition. A branch cut is a line or curve that is removed from thecomplex plane to define a branch of a multiple valued function. Abranch point is a point that is on all branch cuts for a particularfunction.
Example. Any ray θ = α is a branch cut for the logarithm function.The point z = 0 is a branch point for the logarithm.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
We Must Be Careful When Dealing With Branches ofMultivalued Functions
Log(i3
)= Log(−i)
= −π
2i
6= 3π
2i
= 3Log(i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
We Must Be Careful When Dealing With Branches ofMultivalued Functions
Log(i3
)
= Log(−i)
= −π
2i
6= 3π
2i
= 3Log(i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
We Must Be Careful When Dealing With Branches ofMultivalued Functions
Log(i3
)= Log(−i)
= −π
2i
6= 3π
2i
= 3Log(i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
We Must Be Careful When Dealing With Branches ofMultivalued Functions
Log(i3
)= Log(−i)
= −π
2i
6= 3π
2i
= 3Log(i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
We Must Be Careful When Dealing With Branches ofMultivalued Functions
Log(i3
)= Log(−i)
= −π
2i
6= 3π
2i
= 3Log(i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
We Must Be Careful When Dealing With Branches ofMultivalued Functions
Log(i3
)= Log(−i)
= −π
2i
6= 3π
2i
= 3Log(i)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we have
log(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2)
in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2)
= log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)
= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)
= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.
0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0
= log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1)
= log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1))
= log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1)
6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi
0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0
= log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1)
= log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1))
= log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1)
=−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For any two complex numbers z1,z2 we havelog(z1z2) = log(z1)+ log(z2) in the sense that if two of the threevalues are specified as single values, then there is a value for the thirdlogarithm so that the identity works.
Proof.
log(z1z2) = log(
r1eiθ1r2eiθ2)
= log(
r1r2ei(θ1+θ2))
= ln(r1r2)+ i(θ1 +θ2 +2πn)= ln(r1)+ i(θ1 +2πk)+ ln(r2)+ i(θ2 +2πm)= log(z1)+ log(z2)
Example.0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) 6= πi+πi0 = log(1) = log((−1)(−1)) = log(−1)+ log(−1) =−πi+πi
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition.
For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z)
and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof.
Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.
Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1:
Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.
Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1
= znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz
= en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z)
= en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z)
= e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z).
This assignment has n possible values forz = reiθ with 0≤ θ < 2π: z
1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π:
z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Proposition. For z a complex number and n an integer we havezn = en log(z) and this identity has no further qualifications attached.
Proof. Induction on n.Base step n = 1: Done earlier.Induction step n→ (n+1):
zn+1 = znz = en log(z)elog(z) = en log(z)+log(z) = e(n+1) log(z).
So we set z1n := e
1n log(z). This assignment has n possible values for
z = reiθ with 0≤ θ < 2π: z1n = r
1n ei( θ
n + 2πkn ) with k = 0, . . . ,n−1.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition.
For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.
Note. This definition is consistent with the definition of real powersfor real numbers (it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z 6= 0 and c a complex number we define zc := ec log(z)
,which is a multiple valued function.
Note. This definition is consistent with the definition of real powersfor real numbers (it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.
Note. This definition is consistent with the definition of real powersfor real numbers (it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.
Note.
This definition is consistent with the definition of real powersfor real numbers (it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.
Note. This definition is consistent with the definition of real powersfor real numbers
(it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.
Note. This definition is consistent with the definition of real powersfor real numbers (it’s an extension)
as well as with the definition ofinteger powers and roots of complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Definition. For z 6= 0 and c a complex number we define zc := ec log(z),which is a multiple valued function.
Note. This definition is consistent with the definition of real powersfor real numbers (it’s an extension) as well as with the definition ofinteger powers and roots of complex numbers.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Example.
(1+ i)4i = e4i log(1+i) = e4i log(√
2ei π4)
= e4i(ln(√
2)+i π
4 +i2πn) = e−π−8πn+2i ln(2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Example.
(1+ i)4i
= e4i log(1+i) = e4i log(√
2ei π4)
= e4i(ln(√
2)+i π
4 +i2πn) = e−π−8πn+2i ln(2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Example.
(1+ i)4i = e4i log(1+i)
= e4i log(√
2ei π4)
= e4i(ln(√
2)+i π
4 +i2πn) = e−π−8πn+2i ln(2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Example.
(1+ i)4i = e4i log(1+i) = e4i log(√
2ei π4)
= e4i(ln(√
2)+i π
4 +i2πn) = e−π−8πn+2i ln(2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Example.
(1+ i)4i = e4i log(1+i) = e4i log(√
2ei π4)
= e4i(ln(√
2)+i π
4 +i2πn)
= e−π−8πn+2i ln(2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Example.
(1+ i)4i = e4i log(1+i) = e4i log(√
2ei π4)
= e4i(ln(√
2)+i π
4 +i2πn) = e−π−8πn+2i ln(2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem.
For z 6= 0 and c a complex number we have1zc = z−c
Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number we have
1zc = z−c
Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number we have1zc = z−c
Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number we have1zc = z−c
Proof.
zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number we have1zc = z−c
Proof. zcz−c
= ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number we have1zc = z−c
Proof. zcz−c = ec log(z)e−c log(z)
= ec log(z)−c log(z) = e0 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number we have1zc = z−c
Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z)
= e0 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number we have1zc = z−c
Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0
= 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number we have1zc = z−c
Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number we have1zc = z−c
Proof. zcz−c = ec log(z)e−c log(z) = ec log(z)−c log(z) = e0 = 1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem.
For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is
zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z)
where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.
The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain
(r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative is
ddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.
ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc
=ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z)
= ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Theorem. For z 6= 0 and c a complex number, the principal value ofzc is zc := ecLog(z) where Log is the principal value of the logarithm.The principal value is single valued on its domain (r > 0,
−π < θ < π) and its derivative isddz
zc = czc−1.
Proof.ddz
zc =ddz
ecLog(z) = ecLog(z)c1z
= czc 1z
= czc−1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Even When Principal Values are Used, We Must Be CarefulWith Identities
(ii)i = (−1)i = ei(iπ) = e−π
(−i)i(−i)i = ei(−i π
2 )ei(−i π
2 ) = eπ
2 eπ
2 = eπ
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Even When Principal Values are Used, We Must Be CarefulWith Identities
(ii)i
= (−1)i = ei(iπ) = e−π
(−i)i(−i)i = ei(−i π
2 )ei(−i π
2 ) = eπ
2 eπ
2 = eπ
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Even When Principal Values are Used, We Must Be CarefulWith Identities
(ii)i = (−1)i
= ei(iπ) = e−π
(−i)i(−i)i = ei(−i π
2 )ei(−i π
2 ) = eπ
2 eπ
2 = eπ
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Even When Principal Values are Used, We Must Be CarefulWith Identities
(ii)i = (−1)i = ei(iπ)
= e−π
(−i)i(−i)i = ei(−i π
2 )ei(−i π
2 ) = eπ
2 eπ
2 = eπ
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Even When Principal Values are Used, We Must Be CarefulWith Identities
(ii)i = (−1)i = ei(iπ) = e−π
(−i)i(−i)i = ei(−i π
2 )ei(−i π
2 ) = eπ
2 eπ
2 = eπ
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Even When Principal Values are Used, We Must Be CarefulWith Identities
(ii)i = (−1)i = ei(iπ) = e−π
(−i)i(−i)i
= ei(−i π
2 )ei(−i π
2 ) = eπ
2 eπ
2 = eπ
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Even When Principal Values are Used, We Must Be CarefulWith Identities
(ii)i = (−1)i = ei(iπ) = e−π
(−i)i(−i)i = ei(−i π
2 )ei(−i π
2 )
= eπ
2 eπ
2 = eπ
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Even When Principal Values are Used, We Must Be CarefulWith Identities
(ii)i = (−1)i = ei(iπ) = e−π
(−i)i(−i)i = ei(−i π
2 )ei(−i π
2 ) = eπ
2 eπ
2
= eπ
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
logo1
Exponential Function Logarithmic Function Branches and Derivatives Identities Complex Exponents
Even When Principal Values are Used, We Must Be CarefulWith Identities
(ii)i = (−1)i = ei(iπ) = e−π
(−i)i(−i)i = ei(−i π
2 )ei(−i π
2 ) = eπ
2 eπ
2 = eπ
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Exponential and Logarithmic Functions
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