\( \nexists \geq ( \geq \text{ is a partial order on }\mathbb{C} ) \)
Function with a complex domain and codomain
\( f : \mathbb{C} \to \mathbb{C} \)
\(f(x+iy) = u(x,y) + iv(x,y)\)
\(u : \mathbb{R}^2 \to \mathbb{R}\) is the real function
\(v : \mathbb{R}^2 \to \mathbb{R}\) is the complex function
\(\log z = \ln |z| + i \arg (z) , z \in \mathbb{C}\)
\(f \text{ is harmonic on }U \iff \forall u \in U [ \nabla^2 f(u) = 0 ] \)
Function that is differentiable within the complex plane, that is, in contrast to \(\mathbb{R}\) where sequences converging from the left and right must have the same limit, \(\mathbb{C}\) requires any convergent sequence in the complex plane to have the same limit.
\( f \text{ is differentiable at }z \iff \exists f'(z) [ f'(z) = \lim_{s \to z} \frac{f(z) - f(s)}{z-s} ]\)
\( f \text{ is holomorphic on }U \subseteq \mathbb{C} \iff \forall u \in U ( f \text{ is differentiable at }u ) \)
\( f \text{ is entire } \iff f \text{ is holomorphic on } \mathbb{C}\)
\( f \text{ is differentiable at }z_0 \iff \)
\(h=h_1 + ih_2 \land h_1 , h_2 \text{ are Riemann integrable on}[b,a] \implies \int^{b}_{a} h(t)dt = \int^{b}_{a} h_1 (t)dt + i\int^{b}_{a} h_2 (t)dt\)
Class of integrals returning scalar quantities, related to weighting a curve in complex plane
Differentiable curve in a complex space
\(\gamma \text{ is a simple contour parametrized by } z : [t_0,t_1] \to \mathbb{C} \iff \)
Line integral where each point of the curve is weighted by multiplication with the complex function's intensity at that point
\(\displaystyle \int_{\gamma} f(z)dz = \int_{t_0}^{t_1} f(z(t))z'(t)dt\)
\( \displaystyle L(\gamma) = \int^{t_1}_{t_0} |z' (t) |dt\)
\( \displaystyle | \int_{\gamma} f(z)dz | \leq L(\gamma) \max_{z\in \gamma} |f(z)| \)
\( f \text{ is continuous on }\text{dom}(f) \land F'=f \implies \int_{\gamma} f(z)dz = F(z(t_1)) -F(z(t_0)) \)
\( \displaystyle \oint_{\gamma} f = 0 \iff \exists F [ \forall z \in \text{dom}(F) (F'=f) ] \iff \int_{\gamma}f \text{ is soley dependent on endpoints}\)
\(\displaystyle \int_{\gamma} f = \sum^{n}_{k=1} \int_{\gamma_k} f\)
\( \gamma = \sum^{n}_{k=1} \gamma_k \)
\( \gamma_k \text{ is a simple contour parametrized by } z_k : [t_{k-1} , t_{k}] \to \mathbb{C} \)
\(f \text{ is holomorphic on } U \land \gamma \subset \text{ is a closed simple contour}\implies\)
\( \displaystyle \oint_{\gamma} f(z)dz = 0\)
See VCPDE
See topology
\(\displaystyle f \text{ is holomorphic on } U \land \gamma \subset U \text{ is a closed simple contour } \implies f(z) = \frac{1}{2\pi i} \oint_{ \gamma} \frac{f(\xi)}{\xi - z} d\xi\)
\(\displaystyle f \text{ is holomorphic on } U \land \gamma \subset U \text{ is a closed simple contour } \implies f^{(n)}(z) = \frac{n!}{2\pi i} \oint_{\gamma} \frac{f(\xi)}{(\xi - z)^{n+1}} d\xi\)
\(\displaystyle f \text{ is holomorphic on }U \implies f \in C^{\infty}(U) \)
\( f \text{ is holomorphic on }U \implies f \text{ is analytic on }U \)
Note that because holomorphic and analytic properties are logically equivalent, they are used as synonyms
\(|f^{(n)}(z)| \leq \frac{n! \max_{\xi \in \gamma}|f(\xi)|}{r^n}\)
\(f \text{ is entire and bounded }\implies \forall z ( f'(z)=0 ) \)
Singularity that has a limit but is undefined; it can easily be removed by defining the function at that point with the limit
\(z_0\text{ is a removable singularity } \iff f(z_0) = \text{ is undefined } \land \lim_{z \to z_)} f(z) =L\)
Singularity with absolute divergence to \(\infty\)
\(z_0 \text{ is a pole of } f \iff \exists \lim_{z \to z_0}|f(z)|= \infty \land U \subseteq \Omega \)
Poles have a property called order
\(z_0 \text{ is an }n\text{-order pole of }f \iff z_0 \text{ is a pole of }f \land \exists U\subseteq \Omega ( z_0 \in U \land \forall z \in U [ g(z)=(z-z_0)^n f(z) ] ) \)
\(f \text{ is holomorphic on } \Omega \land z_0 \text{ is an isolated zero of }f \implies \)
\(z_0 \text{ is simple pole of } f \iff z_0 \text{ is a } 1\text{-order pole of } f \)
Function that is holomorphic at any non-pole on some set
\(f \text{ is meromorphic on }\Omega \iff \forall z \in \Omega ( \exists U [ f \text{ is holomorphic } \lor \frac{1}{f} \text{ is holomorphic}] ) \)
Infinite series representation of a complex function with the form of a power series, but negative integer exponents. the positive exponents are holomorphic and the negative exponents are meromorphic
\(f(z) = \sum^{\infty}_{n=-\infty} a_{n} (z-c)^n \)
\(f \text{ is meromorphic on }U \implies \exists (a_n)_{n=-\infty}^{\infty} [ \forall z \in U [ f(z) = \sum^{\infty}_{n=-\infty} a_{n} (z-c)^n ] ] \)
Quantity relating to closed integrals around a particular pole
\(\text{Res}(f,c)= a_{-1} \)
\( c \text{ is a pole of }f \text{ with order }n \)
\(\text{Res}(f,c)= \lim_{z \to c} \frac{1}{(n-1)!} \frac{d^{n-1}}{dz^{n-1}} [ (z-c)^n f(z) ] \)
Theorem asserting that the integral of a closed contour integral equals the sum of the residues of all of its interior poles, multiplied by \(2\pi i\)
\(\displaystyle \oint_{\gamma} f(z)dz = 2\pi i \text{Res}(f,c)\)
\(\displaystyle \oint_{\gamma} f(z)dz = 2\pi i \sum^{k}_{i=1}\text{Res}(f,c_i)\)
By taking a path through the complex plane, some real trigonometric integrals can be more easily evaluated through the residue theorem
\(\displaystyle \int^{2\pi}_{0} f( \cos t , \sin t )dt\)
\(z(t)=e^{it}, t \in [0,2\pi] \)
\(dt=\frac{dz}{iz}\)
\( f \text{ is meromorphic on } \{z : \Im (z) \geq 0\} \)
\( \nexists c \in \mathbb{R} (c \text{ is a pole of }f) \)
\( \exists a \in \mathbb{R}\exists R_0 \in \mathbb{R} [ \forall t \in [0,\pi] \forall R \geq R_0 \forall k \geq 2 [ |f(Re^{it})| \leq \frac{a}{R^k} ] \)
\( \displaystyle \implies \int^{\infty}_{-\infty}f(x)dx = 2\pi i \sum^{n}_{k=1} \text{Res}(f,c_k) \)
\( f \text{ is meromorphic on } \{z : \Im (z) \geq 0\} \)
\( \nexists c \in \mathbb{R} (c \text{ is a pole of }f) \)
\( \exists a \in \mathbb{R}\exists R_0 \in \mathbb{R} [ \forall t \in [0,\pi] \forall R \geq R_0 \forall k \geq 2 [ |f(Re^{it})| \leq \frac{a}{|Re^{it}|} ] \)
\(\displaystyle \implies \int^{\infty}_{-\infty}f(x)e^{iyx}dx = 2\pi i \sum^{n}_{k=1} \text{Res}(f,c_k) \)
\(f \text{ is meromorphic on } \Omega \)
\( \gamma \text{ is a simple closed square contour containing all poles of }f \)
\( \forall n \in \mathbb{Z} [ f \text{ is defined at } n ] \)
\( \sum^{\infty}_{n=-\infty} f(n) \text{ converges } \)
\(\displaystyle \implies \sum^{\infty}_{n=-\infty} f(n) =-\sum^{n}_{k=1} \text{Res}(\pi \cot ( \pi z ) f(z),c_k)\)