M3M6: Methods of Mathematical Physics

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

Website: https://github.com/dlfivefifty/M3M6LectureNotes

Lecture 23: Cauchy transforms on the real line

Last lecture we saw that integral equation $$ \lambda u(x) + \int_{0}^\infty K(x-t)u(t) \dt = f(x)\qqfor 0 < x < \infty. $$ could be transformmed into a Riemann–Hilbert problem: find $\Phi(z)$ analytic off $\R$ such that $$ \Phi_+(s) - g(s)\Phi_-(s) = h(s) \qqand \Phi(\infty) = C $$ where $\Phi_\pm(s) = \lim_{\epsilon \rightarrow 0} \Phi(s \pm \I \epsilon)$ are the limits from above and below.

We now wish to study the solution of this equations, using the Plemelj theorem. But to do this we need to discuss Cauchy transforms on the real line

Outline:

1. Cauchy transforms on the Real line
2. Application: Calculating error functions

Cauchy transforms on the real line

Not surprisingly, the key tool to solving the Riemann–Hilberrt problem is going to be the Cauchy transform over the real line: $$ \CC_\R f(z) = {1 \over 2 \pi \I}\int_{-\infty}^\infty {f(s) \over s-z} \D s $$

This is no longer analytic at $\infty$ because the contour of integration passes through there, but we still have Plemelj's theorem under reasonable assumptions:

Theorem (Plemelj on the real line) Assume that $f : \R \rightarrow \C$ is continuously-differentiable and both it and its derivatives are absolutely integrable. Then $\CC f(z)$ is the unique function satisfying the following properties:

1. $\CC f(z)$ is analytic off $\R$
2. $\lim_{z\rightarrow \infty} \CC f(z) = 0$ in any direction
3. $\CC^+ f(x) - \CC^- f(x) = f(x)$ for $x \in \R$

Application: Calculating error functions

Consider the complementary Error function $$ \erfc z = {2 \over \sqrt \pi} \int_z^\infty \E^{-\zeta^2} \D \zeta. $$ We will show that this special function can be reduced to a Cauchy transform over the real line. This function is particularly important in statistics: on the real axis $1-\erfc x$ gives the cummulative density function (CDF) of the Gaussian distribution:

$$\int_{-\infty}^x {\E^{-x^2} \over \sqrt \pi} \dx = 1-{\erfc x\over 2}$$

Note that $\erfc z$ is entire (it's complex-derivative is of course ${2 \over \sqrt \pi} \E^{-z^2}$) but is not nice at $\infty$:

But integrating by parts we see that the dominant behaviour is like $\E^{-z^2}$, $$ \int_z^\infty \E^{-\zeta^2} \D \zeta = \int_z^\infty -2\zeta {\E^{-\zeta^2} \over -2 \zeta} \D \zeta = {\E^{-z^2} \over 2 z} -\int_z^\infty {\E^{-\zeta^2} \over 2 \zeta^2} \D \zeta $$ and indeed $\E^{z^2} \erfc z$ decays in the right-half plane:

Note that $$ {2 \over \sqrt\pi} \int_{-\infty}^z \E^{-\zeta^2} \D \zeta = {2 \over \sqrt\pi} \left[\int_{-\infty}^\infty - \int_z^\infty\right] \E^{-\zeta^2} \D \zeta = 2 - \erfc(z) $$ and has similar properties in the left-hand plane:

Now consider the function $$ \phi(z) = \begin{cases} \E^{z^2}(2-\erfc( z) )& \Re z < 0 \\ -\E^{z^2} \erfc(z) & \Re z > 0 \end{cases} $$ This decays everywhere, and has the jumps for $s$ on the imaginary axis: $$ \phi_+(s) - \phi_-(s) = 2\E^{s^2} $$ where we have oriented from $-\I\infty$ to $\I \infty$, hence $\phi_+(s) = \lim_{\epsilon\rightarrow 0} \phi(s-\epsilon)$ and $\phi_-(s) = \lim_{\epsilon\rightarrow 0} \phi(s+\epsilon)$.

Thus by Plemelj, we have $$ \phi(z) = 2\CC_{(-\I\infty,\I\infty)}[\E^{\diamond^2}](z) $$ or by rotating to the real axis, $$ \phi(z) = 2\CC_{(-\infty,\infty)}[\E^{-\diamond^2}](-\I z). $$ In other words, $$ \erfc z = -2 \E^{-z^2}\CC_{(-\infty,\infty)}[\E^{-\diamond^2}](-\I z) + \begin{cases} 2 & \Re z < 0 \\ 0 & \Re z > 0 \end{cases} $$