$$ \def\dashint{{\int\!\!\!\!\!\!-\,}} \def\infdashint{\dashint_{\!\!\!-\infty}^{\,\infty}} \def\D{\,{\rm d}} \def\dx{\D x} \def\dt{\D t} \def\C{{\mathbb C}} \def\CC{{\cal C}} \def\HH{{\cal H}} \def\I{{\rm i}} \def\qqfor{\qquad\hbox{for}\qquad} $$

M3M6: Methods of Mathematical Physics

Dr. Sheehan Olver
s.olver@imperial.ac.uk

Lecture 2: Cauchy's theorem

References:

1. M.J. Ablowitz & A.S. Fokas, Complex Variables: Introduction and Applications, Second Edition, Cambridge University Press, 2003
2. R. Earl, Metric Spaces and Complex Analysis, https://courses.maths.ox.ac.uk/node/view_material/5392, 2015

Complex-differentiable functions

Definition (Complex-differentiable) Let $D \subset {\mathbb C}$ be an open set. A function $f : D \rightarrow {\mathbb C}$ is called complex-differentiable at a point $z_0 \in D$ if $$ f'(z_0) = \lim_{z \rightarrow z_0} {f(z) - f(z_0) \over z - z_0} $$ exists, for any angle of approach to $z_0$.

Holomorphic functions

Definition (Holomorphic) Let $D \subset {\mathbb C}$ be an open set. A function $f : D \rightarrow {\mathbb C}$ is called holomorphic in $D$ if it is complex-differentiable at all $z \in D$.

Definition (Entire) A function is entire if it is holomorphic in ${\mathbb C}$

Examples

1. $1$ is entire
2. $z$ is entire
3. $1/z$ is holomorphic in ${\mathbb C} \backslash \{0\}$
4. $\sin z$ is entire
5. $\csc z$ is holomorphic in ${\mathbb C} \backslash \{\ldots,-2\pi,-\pi,0,\pi,2\pi,\ldots\}$
6. $\sqrt z$ is holomorphic in ${\mathbb C} \backslash (-\infty,0]$

We can usually infer the domain where a function is holomorphic from a phase portrait, here we see that ${\rm arcsinh}\, z$ has cuts on $[\I,\I \infty)$ and $[-\I,-\I \infty)$, and a zero (red–green–blue–red) at zero, hence we can infer that it is holomorphic in $\C \backslash ([\I,\I \infty) \cup [-\I,-\I \infty))$.

The following example $\sqrt{z-1} \sqrt{z+1}$ is analytic in ${\mathbb C}\backslash [-1,1]$ and will be returned to:

Contours

Definition (Contour) A contour is a continuous & piecewise-continuously differentiable function $\gamma : [a,b] \rightarrow {\mathbb C}$.

Definition (Simple) A simple contour is a contour that is 1-to-1.

Definition (Closed) A closed contour is a contour such that $\gamma(a) = \gamma(b)$

Examples of contours

1. Line segment $[a,b]$ is a simple contour, with $\gamma(t) = t$
2. Arc from $re^{ia}$ to $re^{ib}$ is a simple contour, with $\gamma(t) = re^{i t}$
3. Circle of radius $r$ is a closed simple contour, with $\gamma(t) = re^{i t}$ and $a = -\pi$, $b = \pi$
4. $\gamma(t) = \cos (t+i)^2$ defines a contour that is not simple or closed
5. $\gamma(t) = e^{i t} + e^{2i t}$ for $[a,b] = [-\pi,\pi]$ defines a contour that is closed but not simple

Here's an example of a closed contour that is not simple:

Contour integrals

Definition (Contour integral) The contour integral over $\gamma$ is defined by $$ \int_\gamma f(z) dz := \int_a^b f(\gamma(t)) \gamma'(t) dt $$

An important property of a contour is its arclength:

Definition (Arclength) The arclength of $\gamma$ is defined as $$ {\cal L}(\gamma) := \int_a^b |\gamma'(t)| dt $$

A very useful result is that we can use the maximum and arclength to bound integrals:

Proposition (ML) Let $f : \gamma \rightarrow {\mathbb C}$ and $$ M = \sup_{z \in \gamma} |f(z)| $$
Then $$ \left|\int_\gamma f(z) dz \right| \leq M {\cal L}(\gamma) $$

Cauchy's theorem

Proposition If $f(z)$ is holomorphic on $\gamma$, then $$\int_\gamma f'(z) dz = f(\gamma(b)) - f(\gamma(a))$$

Theorem (Cauchy) If $f$ is holomorphic inside and on a closed contour $\gamma$, then $$\oint_\gamma f(z) dz = 0$$