A complex function is holomorphic or equivalently _[analytic](Analytic%20function.md)_ if it is [differentiable](Complex%20analysis%20(index).md#Derivatives%20of%20Complex%20functions) at every point within its domain. # Singularities A [singularity](Singularity.md) is a point $z_0\in\mathbb{C}$ where $f$ is _not_ holomorphic. You'll find that most functions aren't _globally holomorphic_. [[Liouville's theorem]] shows us why that's the case. ([...see more](Singularity.md#Singularities%20in%20complex%20analysis)) %%It's stated on wikipedia that the term analytic is more general and holomorphic is specific to complex functions and that this is something that had to be proven. look into this!%% # Proving that a function is holomorphic Below are common techniques that, on their own, may each prove that a function is [holomorphic.](Holomorphic%20functions.md) %%This section may need to be redone as an enumerated list of "if and only if" properties.%% ## Proving a function is holomorphic using the Cauchy-Riemann differential equations Consider that:  Its derivative can be taken either in terms of $y$ or in terms of $x$ (check this by plugging in $f'(z)=f'(x+iy)$ into the definition of derivatives). Thus we see that $\partial_x u + i\partial_x v =-i\partial_y u + \partial_y v$ and thus the condition that states complex differentiability can be expressed as a pair of [[1st order PDE]]s $\partial_x u(x,y) = \partial_y v(x,y), \;\;\;\;\; \partial_y u(x,y) = -\partial_x v(x,y).$ Thus, a function is [holomorphic](Holomorphic%20functions.md) if both differential equations hold for a given domain. %%Compare this to the condition that needs to be met for differentiability of real multivariable functions- where in the real case the partial derivatives merely have to exist.%% ## Proving a function is holomorphic by taking its complex Taylor series Equivalently, a function,$f(z)$, is [analytic](Holomorphic%20functions.md) within a radius, $\rho = |z-z_0|$ if it can be expanded into a Taylor series, $\sum_0^{\infty}\frac{f^(z_0)}{n!}(z-z_0)^n$,about $z_0$. $\rho$ is a _radius of convergence_ where $f(z)$ is centered on $z_0$ and the this radius forms a disk lying within some domain $U$ where $f(z)$ is analytic.  # Plotting holomorphic functions The image of complex curves on a 2D plot is a _[conformal](conformal%20map)_ i.e. angle preserving map on the domain in which a function $f(z)$ is holomorphic. %%This may be better expressed as another method of proving that a function is holomorphic.%% --- # Proofs and examples ## Examples of holomorphic and non-holomorphic functions ### Example 1 _globally holomorphic_ functions: $e^z$, $\sin(z)$, $\cos(z)$ can be Taylor expanded within an arbitrarily large convergence radius. ### Example 2 not holomorphic: $z^* = x-iy$, $|z|$. They violate the pair of PDEs in the definition. ### Example 3 holomorphic in $\mathbb{C}-\{w\}$: $\frac{1}{z-w}$ #MathematicalFoundations/Analysis/ComplexAnalysis/Functions