Complex Analysis – Lecture Notes_09-01-2014

Complex Analysis – Lecture Notes BY

DISHANT PANDYA

LECTURER,

DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE,

SCHOOL OF TECHNOLOGY,

PANDIT DEENDAYAL PETROLEUM UNIVERSITY,

RAISAN, GANDHINAGAR, INDIA

dishant.pandya@sot.pdpu.ac.in

Important Statement:-

Please Note that, the main intention behind providing this PPT to all students is to get motivated towards the subject and work in a meticulous manner over the subject to get good score.

The author is not responsible for any misprint or misconception which may arises during the study. You are always & warmly welcome to discuss any relevant topic.

This presentation will give you brief ideas / conceptual understanding of topics viz., Harmonic Functions (H.W.), Harmonic Conjugate Functions (H.W.), Applications of C-R Equations, Applications of Laplace’s Equation, Orthogonal Families of an Analytic function, Many Types of Elementary Functions including Complex Exponential Function, Complex Logarithmic Function, Principal Value of Logarithmic Functions, Periodicity of Exponential Function, Concept of Hyperbolic Functions and consisting of many solved problems for your proper understanding….!

Harmonic Functions:-

We shall see that when a complex function is analytic at a point z, …….

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Harmonic Conjugate Functions:-

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Applications:-

We saw up till now that if the function is analytic in a domain D, then the real and imaginary parts of f are harmonic; that is, both and have continuous second-order partial derivatives and satisfy Laplace’s equation in D:

and . Conversely, if we know that a function is harmonic in D, we can

find a unique (up to an additive constant) harmonic conjugate and construct a function that is analytic in D.

Elementary Functions:-

An alternate description of the definition can be given by:-

Note that, we can also obtain the Principal Value by substituting n = 0 in the general solution.

Moreover,

Thus,

(a).

(b).

Some Important Formulae:-

More Important Formulae:-

Thank You