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