Complex Complex VariablesVariables

ForFor

ECON 397 MacroeconometricsECON 397 Macroeconometrics

Steve CunninghamSteve Cunningham

Open Disks or Open Disks or NeighborhoodsNeighborhoods

Definition. The set of all points z which Definition. The set of all points z which satisfy the inequality |satisfy the inequality |z – zz – z00|<|<, where , where is is a positive real number is called an a positive real number is called an open open diskdisk or or neighborhood of zneighborhood of z0 0 . .

Remark. The Remark. The unit diskunit disk, i.e., the , i.e., the neighborhood |neighborhood |zz|< 1, is of particular |< 1, is of particular significance. significance.

1

Interior PointInterior Point

Definition. A point is called an Definition. A point is called an interior point of S interior point of S if and only if there if and only if there exists at least one neighborhood of exists at least one neighborhood of zz00 which is completely contained in which is completely contained in SS. .

Sz 0

z0

S

Open Set. Closed Set.Open Set. Closed Set.

Definition. If every point of a set Definition. If every point of a set SS is an interior is an interior point of point of SS, we say that , we say that SS is an is an open setopen set. .

Definition. If Definition. If B(S) B(S) S, S, i.e., if i.e., if SS contains all of contains all of its boundary points, then it is called a its boundary points, then it is called a closed closed setset. .

Sets may be neither open nor closed.Sets may be neither open nor closed.

Open ClosedNeither

ConnectedConnected

An open set An open set SS is said to be is said to be connected connected if if every pair of points every pair of points zz11 andand z z22 in in S S can be can be joined by a polygonal line that lies entirely joined by a polygonal line that lies entirely in in SS. Roughly speaking, this means that . Roughly speaking, this means that SS consists of a “single piece”, although it consists of a “single piece”, although it may contain holes. may contain holes.

SS

z1 z2

Domain, Region, Closure, Domain, Region, Closure, Bounded, CompactBounded, Compact

An open, connected set is called a An open, connected set is called a domain. domain. A A regionregion is a domain together with some, none, is a domain together with some, none, or all of its boundary points. The or all of its boundary points. The closure closure of a of a set set SS denoted , is the set of denoted , is the set of SS together with together with all of its boundary. Thusall of its boundary. Thus . .

A set of points A set of points SS is is bounded bounded if there exists a if there exists a positive real number positive real number RR such that | such that |zz|<|<RR for for every every z z S. S.

A region which is both closed and bounded is A region which is both closed and bounded is said to be said to be compact.compact.

S

)(SBSS

Review: Real Functions of Real Review: Real Functions of Real VariablesVariables

Definition. Let Definition. Let . A . A function ffunction f is a rule is a rule which assigns to each element which assigns to each element aa one one and only one element and only one element b b , , . We . We write write f: f: , or in the specific case , or in the specific case b = b = f(a)f(a), and call , and call b “b “the image of the image of aa under under ff.” .” We call We call “the domain of definition of “the domain of definition of f ” f ” or or simply “the domain of simply “the domain of f ”. f ”. We call We call “the “the range of range of f.f.” ” We call the set of all the We call the set of all the imagesimages of of , , denoted denoted f f ((), the image of the function ), the image of the function ff . . We alternately call We alternately call f f a a mapping mapping from from to to ..

Real FunctionReal Function

In effect, a function of a real variable In effect, a function of a real variable maps from one real line to another.maps from one real line to another.

f

Complex FunctionComplex Function

Definition. Complex function of a complex Definition. Complex function of a complex variable. Let variable. Let C. A C. A function f function f defined on defined on is a rule which assigns to each is a rule which assigns to each zz a a complex number complex number w. w. The number The number ww is is called a called a value of f at z value of f at z and is denoted by and is denoted by f(z)f(z), i.e., , i.e., w = f(z). w = f(z). The set The set is called the is called the domain of definition domain of definition of f. of f. Although the domain of definition is Although the domain of definition is often a domain, it need not be.often a domain, it need not be.

RemarkRemark Properties of a real-valued function of a real Properties of a real-valued function of a real

variable are often exhibited by the graph of variable are often exhibited by the graph of the function. But when the function. But when w = f(z)w = f(z), where , where zz and and ww are complex, no such convenient graphical are complex, no such convenient graphical representation is available because each of representation is available because each of the numbers the numbers zz and and ww is located in a plane is located in a plane rather than a line. rather than a line.

We can display some information about the We can display some information about the function by indicating pairs of corresponding function by indicating pairs of corresponding points points z = z = ((x,yx,y) and ) and w = w = ((u,vu,v). To do this, it is ). To do this, it is usually easiest to draw the usually easiest to draw the z z and and w w planes planes separately.separately.

Graph of Complex Graph of Complex FunctionFunction

x u

y v

z-plane

w-plane

domain ofdefinition

range

w = f(z)

Example 1Example 1

Describe the range of the function f(z) = x2 + 2i, defined on (the domain is) the unit disk |z| 1.

Solution: We have u(x,y) = x2 and v(x,y) = 2. Thus as z varies over the closed unit disk, u varies between 0 and 1, and v is constant (=2).

Therefore w = f(z) = u(x,y) + iv(x,y) = x2 +2i is a line segment from w = 2i to w = 1 + 2i. x

y

u

vf(z)

domain

range

Example 2Example 2Describe the function f(z) = z3 for z in the semidisk given by |z| 2, Im z 0.

Solution: We know that the points in the sector of the semidisk from Arg z = 0 to Arg z = 2/3, when cubed cover the entire disk |w| 8 because

The cubes of the remaining points of z also fall into this disk, overlapping it in the upper half-plane as depicted on the next screen.

iiee 2

33

282

2-2 x

y

u

v

8-8

-8

8

2

w = z3

SequenceSequence Definition. A Definition. A sequence of complex numberssequence of complex numbers, ,

denoted denoted , is a function , is a function ff, such that , such that f: f: N N C, i.e, it is a C, i.e, it is a function whose domain is the set of natural function whose domain is the set of natural numbers between 1 and numbers between 1 and kk, and whose range is a , and whose range is a subset of the complex numbers. If subset of the complex numbers. If k = k = , then the , then the sequence is called sequence is called infinite infinite and is denoted by and is denoted by , or more often, , or more often, zzn n .. (The notation (The notation f(n) f(n) is is equivalent.)equivalent.)

Having defined sequences and a means for Having defined sequences and a means for measuring the distance between points, we measuring the distance between points, we proceed to define the limit of a sequence.proceed to define the limit of a sequence.

knz 1

1nz

Limit of a SequenceLimit of a Sequence

Definition. A sequence of complex numbers Definition. A sequence of complex numbers is said to have the limit is said to have the limit zz0 0 , , or to converge or to converge to to zz00 , if for any , if for any > 0, there exists an integer > 0, there exists an integer NN such that | such that |zznn – z – z00| < | < for all for all nn > > NN. We . We denote this by denote this by

Geometrically, this amounts to the fact that Geometrically, this amounts to the fact that zz00 is the only point of is the only point of zznn such that any such that any neighborhood about it, no matter how small, neighborhood about it, no matter how small, contains an infinite number of points contains an infinite number of points zznn . .

1nz

.

lim

0

0

naszz

orzz

n

nn

Limit of a FunctionLimit of a Function

We say that the complex number We say that the complex number ww00 is the limit is the limit of the function of the function f(z) f(z) as as z z approaches approaches zz00 if if f(z) f(z) stays close to stays close to ww00 whenever whenever zz is sufficiently near is sufficiently near zz00 . Formally, we state: . Formally, we state:

Definition. Limit of a Complex Sequence. Let Definition. Limit of a Complex Sequence. Let f(z) f(z) be a function defined in some neighborhood of be a function defined in some neighborhood of zz00 except with the possible exception of the except with the possible exception of the point point zz00 is the number is the number ww00 if for any real number if for any real number > 0 there exists a positive real number > 0 there exists a positive real number > 0 > 0 such that |such that |f(z) – wf(z) – w00|< |< whenever 0<| whenever 0<|zz - - zz00|< |< ..

Limits: InterpretationLimits: Interpretation

We can interpret this to mean that if we observe points z within a radius of of zz00, we , we can find a corresponding disk about can find a corresponding disk about ww00 such that all the points in the disk about such that all the points in the disk about zz00 are mapped into it. That is, are mapped into it. That is, any any neighborhood of neighborhood of ww00 contains all the values contains all the values assumed by assumed by ff in some full neighborhood of in some full neighborhood of zz00, except possibly , except possibly f(zf(z00)). .

zz00 w

0

w = f(z)

z-plane w-planeu

v

x

y

Properties of LimitsProperties of Limits

If as If as zz zz00, lim , lim f(z)f(z) A and lim A and lim g(z)g(z) B, thenB, then

lim [ lim [ f(z) f(z) g(z) g(z) ] = A ] = A B B lim lim f(z)g(z)f(z)g(z) = AB, and = AB, and lim lim f(z)/g(z) f(z)/g(z) = A/B. if B = A/B. if B 0. 0.

ContinuityContinuity

Definition. Let Definition. Let f(z) f(z) be a function such be a function such that that f: C C. f: C C. We call We call f(z) continuous at zf(z) continuous at z00 iff:iff: FF is defined in a neighborhood of z is defined in a neighborhood of z00,, The limit exists, andThe limit exists, and

A function A function f f is said to be is said to be continuous on a continuous on a set Sset S if it is continuous at each point of S. if it is continuous at each point of S. If a function is not continuous at a point, If a function is not continuous at a point, then it is said to be singular at the point.then it is said to be singular at the point.

)()(lim 00

zfzfzz

Note on ContinuityNote on Continuity

One can show that One can show that f(z)f(z) approaches a approaches a limit precisely when its real and limit precisely when its real and imaginary parts approach limits, and imaginary parts approach limits, and the continuity of the continuity of f(z) f(z) is equivalent to is equivalent to the continuity of its real and the continuity of its real and imaginary parts.imaginary parts.

Properties of Continuous Properties of Continuous FunctionsFunctions

If If f(z) f(z) and and g(z) g(z) are continuous at are continuous at zz00, , then so are then so are f(z) f(z) g(z) g(z) and and f(z)g(z). f(z)g(z). The quotient The quotient f(z)/g(z)f(z)/g(z) is also is also continuous at continuous at zz00 provided that provided that g(zg(z00) ) 0. 0.

Also, continuous functions map Also, continuous functions map compact sets into compact sets. compact sets into compact sets.

DerivativesDerivatives

Differentiation of complex-valued functions is Differentiation of complex-valued functions is completely analogous to the real case:completely analogous to the real case:

Definition. Derivative. Let Definition. Derivative. Let f(z) f(z) be a complex-be a complex-valued function defined in a neighborhood of valued function defined in a neighborhood of zz00. Then the . Then the derivative derivative of of f(z) f(z) at at zz00 is given byis given by

Provided this limit exists. Provided this limit exists. F(z) F(z) is said to be is said to be differentiable at zdifferentiable at z00..

zzfzzf

zfz

)()(lim)( 00

00

Properties of DerivativesProperties of Derivatives

Rule. Chain''

.0if,''

'

'''

. constant anyfor ''

'''

000

020

00000

00000

00

000

zgzgfzgfdzd

zgzg

zgzfzfzgz

gf

zgzfzgzfzfg

czcfzcf

zgzfzgf

Analytic. Holomorphic.Analytic. Holomorphic. Definition. A complex-valued function Definition. A complex-valued function ff ( (zz) is ) is

said to be said to be analyticanalytic, or equivalently, , or equivalently, holomorphicholomorphic, on an open set , on an open set if it has a if it has a derivative at every point of derivative at every point of . (The term . (The term “regular” is also used.)“regular” is also used.)

It is important that a function may be It is important that a function may be differentiable at a single point only. Analyticity differentiable at a single point only. Analyticity implies differentiability within a neighborhood implies differentiability within a neighborhood of the point. This permits expansion of the of the point. This permits expansion of the function by a Taylor series about the point.function by a Taylor series about the point.

If If ff ( (zz) is analytic on the whole complex plane, ) is analytic on the whole complex plane, then it is said to be an then it is said to be an entire functionentire function. .

Rational Function.Rational Function.

Definition. If Definition. If ff and and gg are polynomials in are polynomials in zz, then , then h h ((zz) = ) = ff ( (zz)/)/gg((zz), ), gg((zz) ) 0 is 0 is called a rational function. called a rational function.

Remarks.Remarks. All polynomial functions of All polynomial functions of zz are entire. are entire. A rational function of A rational function of zz is analytic at every is analytic at every

point for which its denominator is point for which its denominator is nonzero.nonzero.

If a function can be reduced to a If a function can be reduced to a polynomial function which does not polynomial function which does not involve , then it is analytic. involve , then it is analytic.

z

Example 1Example 1

11

11

)(

21

2

21

2)(

2,

2

)1(1

)(

1

221

221

zzzzzz

zf

izzzz

izz

izz

zf

izz

yzz

xlet

yxiyx

zf

Thus f1(z) is analytic at all points except z=1.

Example 2Example 2

13)(2

312

322

)(

2,

2

313)(

2

22

2

222

zzzzfizz

izz

izzzz

zf

izz

yzz

xlet

yixyxzf

Thus f2(z) is nowhere analytic.

Testing for AnalyticityTesting for Analyticity

Determining the analyticity of a function by searching for in its expression that cannot be removed is at best awkward. Observe:

z

5523

5345 1)(

zzzz

zzzzzzzf

It would be difficult and time consuming to try to reduce this expression to a form in which you could be sure that the could not be removed. The method cannot be used when anything but algebraic functions are used.

z

Cauchy-Riemann Cauchy-Riemann Equations (1)Equations (1)

If the function f (z) = u(x,y) + iv(x,y) is differentiable at z0 = x0 + iy0, then the limit

zzfzzf

zfz

)()(lim)( 00

00

can be evaluated by allowing z to approach zero from any direction in the complex plane.

Cauchy-Riemann Cauchy-Riemann Equations (2)Equations (2)

If it approaches along the x-axis, then z = x, and we obtain

x

yxivyxuyxxivyxxuzf

x

),(),(),(),(lim)(' 00000000

00

xyxvyxxv

ix

yxuyxxuzf

xx

),(),(lim

),(),(lim)(' 0000

0

0000

00

But the limits of the bracketed expression are just the first partial derivatives of u and v with respect to x, so that:

).,(),()(' 00000 yxxv

iyxxu

zf

Cauchy-Riemann Cauchy-Riemann Equations (3)Equations (3)

If it approaches along the y-axis, then z = y, and we obtain

y

yxuyyxuzf

y

),(),(lim)(' 0000

00

And, therefore

).,(),()(' 00000 yxyv

yxyu

izf

y

yxvyyxvi

y

),(),(lim 0000

0

Cauchy-Riemann Cauchy-Riemann Equations (4)Equations (4)

By definition, a limit exists only if it is unique. Therefore, these two expressions must be equivalent. Equating real and imaginary parts, we have that

xv

yu

yv

xu

and

must hold at z0 = x0 + iy0 . These equations are called the Cauchy-Riemann Equations. Their importance is made clear in the following theorem.

Cauchy-Riemann Cauchy-Riemann Equations (5)Equations (5)

Theorem. Let Theorem. Let ff ( (zz) = ) = uu((x,yx,y) + ) + iviv((x,yx,y) be ) be defined in some open set defined in some open set containing containing the point the point zz00. If the first partial . If the first partial derivatives of derivatives of u u and and vv exist in exist in , and are , and are continuous at continuous at zz0 0 , and satisfy the , and satisfy the Cauchy-Riemann equations at Cauchy-Riemann equations at zz00, then , then ff ((zz) is differentiable at ) is differentiable at zz00. Consequently, . Consequently, if the first partial derivatives are if the first partial derivatives are continuous and satisfy the Cauchy-continuous and satisfy the Cauchy-Riemann equations at Riemann equations at all all points of points of , , then then ff ( (zz) is ) is analytic in analytic in ..

Example 1Example 1

1,1,2,2

)()()( 22

xv

yu

yyv

xxu

xyiyxzf

Hence, the Cauchy-Riemann equations are satisfied only on the line x = y, and therefore in no open disk. Thus, by the theorem, f (z) is nowhere analytic.

Example 2Example 2Prove that f (z) is entire and find its derivative.

yexv

yeyu

yeyv

yexu

yieyezf

xxxx

xx

sin,sin,cos,cos

:Solution

sincos)(

The first partials are continuous and satisfy the Cauchy-Riemann equations at every point.

.sincos)(' yieyexv

ixu

zf xx

Harmonic FunctionsHarmonic Functions

Definition. Harmonic. A real-valued Definition. Harmonic. A real-valued function function ((xx,,yy) is said to be ) is said to be harmonic harmonic in a in a domain domain DD if all of its second-order partial if all of its second-order partial derivatives are continuous in derivatives are continuous in DD and if each and if each point of point of DD satisfies satisfies

.02

2

2

2

yx

Theorem. If f (z) = u(x,y) + iv(x,y) is analytic in a domain D, then each of the functions u(x,y) and v(x,y) is harmonic in D.

Harmonic ConjugateHarmonic Conjugate

Given a function Given a function uu((xx,,yy) harmonic in, ) harmonic in, say, an open disk, then we can find say, an open disk, then we can find another harmonic function another harmonic function vv((xx,,yy) so ) so that that u + ivu + iv is an analytic function of is an analytic function of zz in the disk. Such a function in the disk. Such a function vv is called is called a a harmonic conjugateharmonic conjugate of of u.u.

ExampleExampleConstruct an analytic function whose real part is: .3),( 23 yxyxyxu

Solution: First verify that this function is harmonic.

.066

6 and16

6and33

2

2

2

2

2

2

2

222

xxyu

xu

and

xyu

xyyu

xxu

yxxu

Example, Example, ContinuedContinued

16)2(

33)1( 22

xyyu

xv

andyxxu

yv

Integrate (1) with respect to y:

)(3),()3(

33

33

32

22

22

xyyxyxv

yyxv

yyxv

Example, Example, ContinuedContinued

Now take the derivative of v(x,y) with respect to x:

).('6 xxyxv

According to equation (2), this equals 6xy – 1. Thus,

.3),(

.)(and,)(So

.1ly,Equivalent.1)('and

16)('6

32 CyyxyxvAnd

Cxxxx

xx

xyxxy

Example, Example, ContinuedContinued

The desired analytic function f (z) = u + iv is:

Cxyyxiyxyxzf 3223 33)(

Complex ExponentialComplex Exponential

We would like the complex exponential We would like the complex exponential to be a natural extension of the real to be a natural extension of the real case, with case, with ff ( (zz) = e) = ezz entire. We begin by entire. We begin by examining eexamining ezz = = eex+iy x+iy = e= exxeeiyiy..

eeiy iy = cos = cos yy + + ii sin sin y y by Euler’s and by Euler’s and DeMoivre’s relations.DeMoivre’s relations.

Definition. Complex Exponential Definition. Complex Exponential Function. If Function. If zz = = xx + + iyiy, then e, then ezz = e = exx(cos (cos yy + + i i sin sin yy).).

That is, |eThat is, |ezz|= e|= exx and arg e and arg ezz = = yy..

More on ExponentialsMore on Exponentials

Recall that a function Recall that a function ff is is one-to-oneone-to-one on a set on a set SS if the equation if the equation ff ( (zz11) = ) = ff ( (zz22), where ), where zz11,, zz2 2 SS, , implies that implies that zz1 1 = = zz22. The complex exponential . The complex exponential function is function is notnot one-to-one on the whole plane. one-to-one on the whole plane.

Theorem. A necessary and sufficient condition Theorem. A necessary and sufficient condition that ethat ezz = 1 is that = 1 is that zz = 2 = 2kkii, where , where kk is an is an integer. integer.

Also, a necessary and sufficient condition that Also, a necessary and sufficient condition that is that is that zz1 1 = = zz2 2 ++ 22kkii, where , where kk is an is an

integer. Thus integer. Thus eezz is a periodic function. is a periodic function.

21 ee zz