List of basic formulas and methods that can help you find functions of a complex variable:

  1. Definition of a function of a complex variable:
  • The function ( f(z) ) is defined as ( f(z) = u(x, y) + iv(x, y) ), where ( z = x + iy ), ( u ) and ( v ) are real functions of the variables ( x ) and ( y ).
  1. Cauchy-Riemann criterion:
  • For a function ( f(z) ) to be analytic in a domain, the following conditions must be satisfied:
    [
    \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
    ]
  1. Contour integral:
  • If ( f(z) ) is analytic in the domain, then the integral over the closed contour is zero:
    [
    \oint_C f(z) \, dz = 0
    ]
  1. Cauchy formula for the integral:
  • If ( f(z) ) is analytic inside and on the contour ( C ):
    [
    f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{za} \, dz
    ]
  1. Taylor series expansion:
  • If ( f(z) ) is analytic at point ( a ):
    [
    f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (za)^n
    ]
  1. Laurent series expansion:
  • For a function that can have singular points:
    [
    f(z) = \sum_{n=-\infty}^{\infty} a_n (za)^n
    ]
  1. Transformation of a complex variable:
  • If (z = re^{i\theta}), then polar coordinates can be used to analyze the function.

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