Subjects: Complex Analysis
Links: Continuous Functions in C, Complex Numbers, The Derivative on R, Important Functions in Complex Numbers

Def: Let $f:A\subseteq \mathbb{C}\to \mathbb{C}$ where $A\subseteq \mathbb{C}$ is an open set. The function $f$ is said to be differentiable (in the comples sense) at $z_0\in A$ if

exists. The limit denoted as $f^{\prime }(z_0)$, or sometimes by ${\displaystyle \frac{df}{dz}}(z_0)$. Thus $f^{\prime }(z_0)$ is a complex number. The function $f$ is said to be ***********holoorphic on $A$ if $f$ is complex differentiable at each $z_0\in A$. The word holomorphic is synominous to analytic. The phrase ****holomorphic at $z_0$ means $f$ is analyitic on a neighborhood of $z_0$.

Prop: If $f^{\prime }(z_0)$ exists, then $f$ is continuous at $z_0$.

Prop: Let $f,g:A\subseteq \mathbb{C}\to \mathbb{C}$ and $z_0\in A$. If $f^{\prime }(z_0)$ and $g^{\prime }(z_0)$ exists then:

• $f+g$ is differentiable at $z_0$ and $(f+g{)}^{\prime }(z_0)=f^{\prime }(z_0)+g^{\prime }(z_0)$
• If $c\in \mathbb{C}$, then $cf$ is differentiable at $z_0$ and $(cf{)}^{\prime }(z_0)=c{f}^{\prime }(z_0)$
• $fg$ is differentiable at $z_0$ and $(fg{)}^{\prime }(z_0)=f^{\prime }(z_0)g(z_0)+f(z_0)g^{\prime }(z_0)$
• If $g(z_0)\ne 0$, then $f/g$ is differentiable at $z_0$ and

Chain Rule

Let $g:\mathrm{\Omega }\subseteq \mathbb{C}\to \mathbb{C}$ and $f:\bar{\mathrm{\Omega }}\subseteq \mathbb{C}\to \mathbb{C}$ , such that $f[\bar{\mathrm{\Omega }}]\subseteq \mathrm{\Omega }$ and $z_0\in \bar{\mathrm{\Omega }}$. If $f$ is differentiable at $z_0$ and $g$ is differentiable at $f(z_0)$, then $g\circ f$ is differentiable at $z_0$ and

Prop: Any polinomial $p\in \mathbb{C}[x]$ is holomorphic in all $\mathbb{C}$, and any rational function with $p,q\in \mathbb{C}[x]$ of the form $p/q$ is analyitic on all $\mathbb{C}$ except on the roots of $q$.

Prop: Let $f$ be holomorphic on an open and connected set $A$, any of the following conditions is enough to conclude that $f$ is constant.

• $|f|$ is a constant on $A$
• $f(z)\in \mathbb{R}$ for all $z\in A$
• $f^{\prime }=0$ on $A$

Prop: The function $\exp :\mathbb{C}\to \mathbb{C}$ is holomorphic on $\mathbb{C}$ and

Cor: The trigonometric functions $\sin ,\cos :\mathbb{C}\to \mathbb{C}$ are holomorphic on $\mathbb{C}$ and

Prop: For any $y_0\in \mathbb{R}$, then the ${\ln }_{y_0}:{\mathbb{C}}^{\ast }\to \mathbb{C}$ is holomorphic on $\mathbb{C}\setminus \{r(\cos y_0+i\sin y_0)\mid r\ge 0\}$ and

for $z\in \mathbb{C}\setminus \{r(\cos y_0+i\sin y_0)\mid r\ge 0\}$.

Cor: Let $g:\mathbb{C}\to A_{\alpha ,n}$, defined as

is the $n$th root with image on $A_{\alpha ,n}$. Then $g$ is continuous and holomorphic at$\mathbb{C}\setminus \{r(\cos (n\alpha )+i\sin (n\alpha ))\mid r\ge 0\}$

Prop: Let $g(z)=z^n$ with $g:\mathbb{C}\to \mathbb{C}$ is holomorphic on $\mathbb{C}$, and

Differentiation of Elementary Functions

Prop: The function $\exp :\mathbb{C}\to \mathbb{C}$ is holomorphic on $\mathbb{C}$ and

Cor: The trigonometric functions $\sin ,\cos :\mathbb{C}\to \mathbb{C}$ are holomorphic on $\mathbb{C}$ and

Prop: For any $y_0\in \mathbb{R}$, then the ${\ln }_{y_0}:{\mathbb{C}}^{\ast }\to \mathbb{C}$ is holomorphic on $\mathbb{C}\setminus \{r(\cos y_0+i\sin y_0)\mid r\ge 0\}$ and

for $z\in \mathbb{C}\setminus \{r(\cos y_0+i\sin y_0)\mid r\ge 0\}$.

Cor: Let $g:\mathbb{C}\to A_{\alpha ,n}$, defined as

is the $n$th root with image on $A_{\alpha ,n}$. Then $g$ is continuous and holomorphic at$\mathbb{C}\setminus \{r(\cos (n\alpha )+i\sin (n\alpha ))\mid r\ge 0\}$

Prop: Let $g(z)=z^n$ with $g:\mathbb{C}\to \mathbb{C}$ is holomorphic on $\mathbb{C}$, and