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Chapter 20 Complex variables

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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)

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Chapter 20 Complex variables

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20.3 Power series in a complex variable

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Chapter 20 Complex variables 20.4 Some elementary functions

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Chapter 20 Complex variables

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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)

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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.

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Chapter 20 Complex variables

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(A) (B)

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Chapter 20 Complex variables 20.6 Singularities and zeros of complex function

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Chapter 20 Complex variables

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20.10 Complex integral

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Chapter 20 Complex variables

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20.11 Cauchy theorem

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Chapter 20 Complex variables

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20.12 Cauchys integral formula

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20.13 Taylor and Laurent series Taylors theorem:

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Chapter 20 Complex variables

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How to obtain the residue ?

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Chapter 20 Complex variables

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20.14 Residue theorem

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Chapter 20 Complex variables Residue theorem:

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Chapter 20 Complex variables

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20.16 Integrals of sinusoidal functions

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20.17 Some infinite integrals

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Chapter 20 Complex variables

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For poles on the real axis:

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Chapter 20 Complex variables Jordans lemma

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Chapter 20 Complex variables

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20.18 Integral of multivalued functions

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Chapter 20 Complex variables

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