fact:

Useful facts: $\begin{align*} f' = {\frac{\partial f}{\partial z}\,} = {1\over i}{\frac{\partial f}{\partial y}\,} = {\frac{\partial f}{\partial x}\,} = {\frac{\partial u}{\partial x}\,} + i {\frac{\partial v}{\partial x}\,} .\end{align*}$

Complex Calculus

fact:

Various differentials: $\begin{align*} dz &= dx + i~dy \\ d\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu &= dx - i~dy \\ \\ f_z &= f_x = f_y / i .\end{align*}$

Integral of a complex exponential: $\begin{align*} \int_{0}^{2 \pi} e^{i \ell x} d x &=\left\{\begin{array}{ll} {2 \pi} & {\ell=0} \\ {0} & \text{else} \end{array}\right. .\end{align*}$

$\begin{align*} \int_{\gamma} f(z) \,dz= \int_0^{2\pi } f(Re^{i\theta}) \, iRe^{i\theta} \,d\theta .\end{align*}$

warnings:

$f(z) = \sin(z), \cos(z)$ are unbounded on ${\mathbb{C}}$ ! An easy way to see this: they are nonconstant and entire, thus unbounded by Liouville.

example (Square root is not holomorphic):

You can show $f(z) = \sqrt{z}$ is not holomorphic by showing its integral over $S^1$ is nonzero. This is a direct computation: $\begin{align*} \int_{S^1} z^{1/2} \,dz &= \int_0^{2\pi} (e^{i\theta})^{1/2} ie^{i\theta} \,d\theta\\ &= i \int_0^{2\pi} e^{i3\theta \over 2}\,d\theta\\ &= i \qty{2\over 3i} e^{i3\theta \over 2}\Big|_{0}^{2\pi} \\ &= {2\over 3}\qty{e^{3\pi i - 1}} \\ &= -{4\over 3} .\end{align*}$

Note an issue: a different parameterization yields a different (still nonzero) number $\begin{align*} \cdots &= \int_{-\pi}^{\pi} (e^{i\theta})^{1/2} ie^{i\theta} \,d\theta\\ &= {2\over 3}\qty{ e^{3\pi i \over 2} - e^{-3\pi i \over 2}} \\ &= -{4i\over 3} .\end{align*}$ This is these are paths that don’t lift to closed loops on the Riemann surface defined by $z\mapsto z^2$ .

remark (Solving real integrals with complex calculus):

The function $e^z$ is entire, so “usual” calculus using it works. For example, $\begin{align*} \int e^{3x}\cos(2x) \,dx &= \Re \int e^{3z}e^{2iz}\,dz\\ &= \Re \int e^{(3+2i)z} \,dz\\ &= \Re {e^{(3+2i)z} \over 3+2i} + C \\ &= e^{2x \over 13}\qty{\cos(2x) + i\sin(2x)} .\end{align*}$

Holomorphy

remark:

Let $f:{\mathbb{C}}\to {\mathbb{C}}$ satisfy CR and consider it as a map $f:{\mathbb{R}}^2\to{\mathbb{R}}^2$ , writing $f(x+iy) = u(x, y) + iv(x, y)$ , the Jacobian is $\begin{align*} J = u_xv_y -v_x u_y = u_x^2 + v_x^2 .\end{align*}$ OTOH, $\begin{align*} {\left\lvert {f'(z)} \right\rvert}^2 = {\left\lvert {u_x + iv_y} \right\rvert}^2 = J .\end{align*}$

definition (Complex differentiable / holomorphic /entire):

A function $f: {\mathbb{C}}\to {\mathbb{C}}$ is complex differentiable or holomorphic at $z_0$ iff the following limit exists: $\begin{align*} \lim_{h\to 0} { f(z_0 + h) - f(h) \over h } .\end{align*}$ A function that is holomorphic on ${\mathbb{C}}$ is said to be entire.

Equivalently, there exists an $\alpha\in {\mathbb{C}}$ such that $\begin{align*} f(z_0+h) - f(z_0) = \alpha h + R(h) && R(h) \overset{h\to 0}\longrightarrow 0 .\end{align*}$ In this case, $\alpha = f'(z_0)$ .

example (Holomorphic vs non-holomorphic):

$f(z) \coloneqq{\left\lvert {z} \right\rvert}$ is not holomorphic.
$f(z) \coloneqq\arg{z}$ is not holomorphic.
$f(z) \coloneqq\Re{z}$ is not holomorphic.
$f(z) \coloneqq\Im{z}$ is not holomorphic.
$f(z) = {1\over z}$ is holomorphic on ${\mathbb{C}}\setminus\left\{{0}\right\}$ but not holomorphic on ${\mathbb{C}}$
$f(z) = \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu$ is not holomorphic, but is real differentiable: $\begin{align*} {f(z_0 + h) - f(z_0) \over h } = {\mkern 1.5mu\overline{\mkern-1.5muz_0\mkern-1.5mu}\mkern 1.5mu + \mkern 1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu - \mkern 1.5mu\overline{\mkern-1.5muz_0\mkern-1.5mu}\mkern 1.5mu \over h} = {\mkern 1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu \over h} = {re^{-i\theta} \over re^{i\theta}} = e^{-2i\theta} \overset{h\to 0}\longrightarrow e^{-2i\theta} ,\end{align*}$ which is a complex number that depends on $\theta$ and is thus not a single value.

definition (Real (multivariate) differentiable):

A function $F: {\mathbb{R}}^n\to {\mathbb{R}}^m$ is real-differentiable at $\mathbf{p}$ iff there exists a linear transformation $A$ such that $\begin{align*} { {\left\lVert { F(\mathbf{p} + \mathbf{h}) - F(\mathbf{p}) - A(\mathbf{h}) } \right\rVert} \over {\left\lVert { \mathbf{h} } \right\rVert} } \overset{{\left\lVert {\mathbf{h}} \right\rVert}\to 0}\longrightarrow 0 .\end{align*}$ Rewriting, $\begin{align*} {\left\lVert { F(\mathbf{p} + \mathbf{h}) - F(\mathbf{p}) - A(\mathbf{h}) } \right\rVert} = {\left\lVert { \mathbf{h} } \right\rVert} {\left\lVert { R(\mathbf{h}) } \right\rVert} && {\left\lVert {R(\mathbf{h}) } \right\rVert}\overset{{\left\lVert {\mathbf{h} } \right\rVert} \to 0}\longrightarrow 0 .\end{align*}$

Equivalently, $\begin{align*} F(\mathbf{p} + \mathbf{h}) - F(\mathbf{p}) = A(\mathbf{h}) + {\left\lVert {\mathbf{h}} \right\rVert} R(\mathbf{h}) && {\left\lVert {R(\mathbf{h}) } \right\rVert}\overset{{\left\lVert {\mathbf{h} } \right\rVert} \to 0}\longrightarrow 0 .\end{align*}$

Or in a slightly more useful form, $\begin{align*} F(\mathbf{p} + \mathbf{h}) = F(\mathbf{p}) + A(\mathbf{h}) + R(\mathbf{h}) && R\in o( {\left\lVert {\mathbf{h}} \right\rVert}), \text{ i.e. } { {\left\lVert { R(\mathbf{h}) } \right\rVert} \over {\left\lVert {\mathbf{h}} \right\rVert}} \overset{\mathbf{h}\to 0}\longrightarrow 0 .\end{align*}$

proposition (Holomorphic implies continuous.):

$f$ is holomorphic at $z_0$ iff there exists an $a\in {\mathbb{C}}$ such that $\begin{align*} f(z_0 + h) - f(z_0) - ah = h \psi(h), \quad \psi(h) \overset{h\to 0}\to 0 .\end{align*}$ In this case, $a = f'(z_0)$ .

#todo proof

Wirtinger Calculus

remark:

Some properties:

$\mkern 1.5mu\overline{\mkern-1.5mu{ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}f(z)\mkern-1.5mu}\mkern 1.5mu = {\partial}\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu(z)$ , or ${\partial}f^*(z) = ({ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}f(z))^*$
- E.g. $d(cz) = c,dz+ 0,d\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu$ and $d(c\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu) = 0,dz+ c,d\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu$
- E.g. ${\partial}{\left\lvert {z} \right\rvert}^2 = {\partial}z\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu = \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu$ and ${ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}{\left\lvert {z} \right\rvert}^2 = z$ .
  - So $d({\left\lvert {z} \right\rvert}^2) = \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu,dz+ z,d\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu$
- E.g. ${\partial}\exp\qty{ - {\left\lvert {z} \right\rvert}^2 } = {\partial}\exp\qty{-z\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu} = e^{-{\left\lvert {z} \right\rvert}^2}\cdot {\partial}(z\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu) = \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu e^{-{\left\lvert {z} \right\rvert}^2}$ .

Exercises

exercise (?):

Show that a real-valued holomorphic function is constant.

#complex/exercise/completed

solution:

Use that ${\mathbb{R}}$ is not open in ${\mathbb{C}}$ and apply the open mapping theorem to conclude: $f({\mathbb{C}})$ must be open in ${\mathbb{C}}$ if $f$ is holomorphic and nonconstant.

exercise (?):

Show that if $fg \equiv 0$ on a domain $\Omega$ then either $f\equiv 0$ or $g\equiv 0$ .

#complex/exercise/completed

solution:

If $f\not\equiv 0$ , there is a point $z_0$ where $f(z_0)\neq 0$ , and thus a neighborhood $U\ni z_0$ where $f$ is nonvanishing. This forces $g\equiv 0$ on $U$ , however $U$ is a set with a limit point, so $g\equiv 0$ on $\Omega$ by the identity principle.

exercise (?):

Find all entire functions $f$ such that $f(x) = e^x$ on ${\mathbb{R}}$ .

#complex/exercise/completed

solution:

The function $g(z) \coloneqq f(z) - e^z$ is entire and identically zero on ${\mathbb{R}}$ , which contains a limit point. So $g(z) \equiv 0$ on ${\mathbb{C}}$ , meaning $f(z) = e^z$ is the only such function.

Holomorphy and Calculus

Complex Calculus

Holomorphy

Wirtinger Calculus

Exercises