# Chapter 20 Complex variables. 20.2 Cauchy-Riemann relation A function f(z)=u(x,y)+iv(x,y) is differentiable and analytic, there must be particular.

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Chapter 20 Complex variables

20.2 Cauchy-Riemann relation A function f(z)=u(x,y)+iv(x,y) is differentiable and analytic, there must be particular connection between u(x,y) and v(x,y)

Chapter 20 Complex variables

20.3 Power series in a complex variable

Chapter 20 Complex variables 20.4 Some elementary functions

Chapter 20 Complex variables

20.5 Multivalued functions and branch cuts A logarithmic function, a complex power and a complex root are all multivalued. Is the properties of analytic function still applied? (A) (B) (A) (B)

Chapter 20 Complex variables Branch point: z remains unchanged while z traverse a closed contour C about some point. But a function f(z) changes after one complete circuit. Branch cut: It is a line (or curve) in the complex plane that we must cross, so the function remains single-valued.

Chapter 20 Complex variables

(A) (B)

Chapter 20 Complex variables 20.6 Singularities and zeros of complex function

Chapter 20 Complex variables

20.10 Complex integral

Chapter 20 Complex variables

20.11 Cauchy theorem

Chapter 20 Complex variables

20.12 Cauchys integral formula

Chapter 20 Complex variables

20.13 Taylor and Laurent series Taylors theorem:

Chapter 20 Complex variables

How to obtain the residue ?

Chapter 20 Complex variables

20.14 Residue theorem

Chapter 20 Complex variables Residue theorem:

Chapter 20 Complex variables

20.16 Integrals of sinusoidal functions

Chapter 20 Complex variables

20.17 Some infinite integrals

Chapter 20 Complex variables

For poles on the real axis:

Chapter 20 Complex variables Jordans lemma

Chapter 20 Complex variables

20.18 Integral of multivalued functions

Chapter 20 Complex variables

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