MTH 324 Lecture 3 Functions of a complex

MTH 324 Lecture # 3 Functions of a complex variable 1

Previous Lecture’s Review • Polar Form of complex number • Powers and roots • Comparison with Real analysis 2

Lecture’s Outline • Complex valued function • Complex exponential function • Polar form of a function • Periodic function 3

Function A function f from a set A to a set B is a rule that assigns to each element in A one and only one element in B. A =Dom ( f ) and Range ( f ) is a subset of B. 4

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Complex valued function: A complex valued function is a function f whose domain and range are subsets of the C of complex numbers. Notation: w = f(z) Example: 7

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Real and imaginary parts of a complex function: Example: Solution: 13

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Complex exponential function Example Solution 16

Complex exponential function Example Solution 17

Periodic function Example Solution 18

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Exponential form of a complex number 20

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Polar form of a complex function Example Solution 22

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Real Valued functions of a complex variable Examples 24

Complex Valued functions of a real variable Examples 25

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Comparison of Real valued functions with complex valued functions • Real valued functions of a real variable and real valued functions of two real variables are special cases of complex valued functions. • Exponential function is periodic in complex but not in real. 30

References • A First Course in Complex Analysis with Applications by Dennis G. Zill and Patrick D. Shanahan. 31