Complex Variables Theorems

The Cauchy-Goursat Theorem

if 𝑓 is analytic on and inside a simple closed contour
𝐶, then

Cauchy’s Integral Theorem

if 𝑓 is analytic on and inside the simple closed contour 𝐶,
then for any 𝑧0 in the interior of the contour,

Corollary to Cauchy’s Integral Theorem

if 𝑓 is analytic on and inside the simple
closed contour 𝐶, then for any 𝑧₀ inside of 𝐶,

Liouville’s Theorem

If f is entire and bounded in the complex plane, then f is constant

Taylor’s Theorom

basically you can expand it normally….kinda

Laurent’s Theorom

If f is analytic on all points inside an annulus about some point z₀, i.e. there exists real numbers R₁ and R₂ such that 0≤ R₁ ≤ | z - z₀ | ∠ R₂