MTH 324 Lecture 32 Review and some applications

MTH 324 Lecture # 32 Review and some applications of complex Analysis 1

Previous Lecture’s Review • Point wise convergence of real sequences • Uniform convergence of real sequence and series • Point wise convergence of complex sequences • Uniform convergence of real sequence and series 2

Lecture’s Outline • Review of complex Analysis • Applications of complex Analysis 3

Analyticity at a point: Analyticity in a domain: 4

Theorem: (L’Hospital Rule) 5

Remark: 6

A necessary condition for analyticity 7

Harmonic function: Remark: 8

Harmonic conjugate functions 9

Harmonic equation in polar form: 10

Level curve: 11

Orthogonal families: Condition for two families of curves to be orthogonal: 12

Example: Solution: 13

14

Complex potential: 15

Complex exponential function: Properties of exponential function 16

Complex logarithmic function: Remark: 17

Logarithmic identities: 18

Complex powers: Remark: 19

Complex trigonometric functions: 20

Complex trigonometric functions: 21

Complex hyperbolic sine and cosine: 22

Complex integral: Remark: 23

Evaluation of contour integral: 24

Properties of contour integral: 25

Cauchy’s theorem: Cauchy-Goursat theorem: 26

Independence of path: Remark: 27

Cauchy’s integral formula: Cauchy’s derivative formula: 28

Taylor series: Laurent’s Theorem: 29

Zeros of a function: Remark: 30

Pole: Remark: 31

Residue: 32

Residue at a simple pole Residue at a pole of order n 33

Cauchy’s residue theorem 34

References • A First Course in Complex Analysis with Applications by Dennis G. Zill and Patrick D. Shanahan. • Complex variables and applications by James Brown and Ruel Churchill • Fundamentals of Complex Analysis by Muhammad Iqbal 35