List of basic formulas and methods that can help you find functions of a complex variable:
- 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 ).
- 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}
]
- 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
]
- 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
]
- Taylor series expansion:
- If ( f(z) ) is analytic at point ( a ):
[
f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (za)^n
]
- Laurent series expansion:
- For a function that can have singular points:
[
f(z) = \sum_{n=-\infty}^{\infty} a_n (za)^n
]
- Transformation of a complex variable:
- If (z = re^{i\theta}), then polar coordinates can be used to analyze the function.