Harnack theorem

Introduction.

It is frequent to make use of Cauchy integral formula to represent analytically some functions that are useful in mathematical physics applications. However, it must be noted that such representation is not unique, so that the same function can be represented by different integrals of Cauchy’s type. An important case has to do with the equality of the two Cauchy integrals

${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_C}{\displaystyle \frac{{\psi }_1(\zeta )}{\zeta -z}}𝑑\zeta ={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_C}{\displaystyle \frac{{\psi }_2(\zeta )}{\zeta -z}}𝑑\zeta$

for all values of $z$ in the interior of $C$. In general no conclusion can be drawn concerning the equality of the density functions ${\psi }_1(\zeta )$ and ${\psi }_2(\zeta )$. We shall see, however, that if some additional restriction are imposed on the density functions and on the contour $C$, then the equality will occur. That is the matter of Harnack theorem. In considering the applications of the theory of functions of a complex variable to problems in continuum mechanics, for instance, we shall most frequently deal with the region bounded by the unit circle, i.e. the compact disc $|z|\le 1$ that we shall draw in the $z$-plane, its boundary will be denoted by $\gamma$ and the points of $\gamma$ by $\zeta =e^{i\theta }$. All density functions of the argument $\theta$ will be assumed to be $2\pi$-periodic.

Theorem 1.

Let $f\mathit{}\mathrm{(}\theta \mathrm{)}$ and $g\mathit{}\mathrm{(}\theta \mathrm{)}$ be continuous real functions of the argument $\theta$ defined on the boundary $\gamma$; if

${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{f(\theta )d\zeta }{\zeta -z}}={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{g(\theta )d\zeta }{\zeta -z}}$ (1)

then

$f(\theta )=g(\theta )+const.\mathit{\hspace{1em}\hspace{1em}}if\mathit{\hspace{1em}\hspace{1em}}|z|>1.$

Proof.

1. . From equality (1) we obtain

   ${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{f(\theta )-g(\theta )}{\zeta -z}}𝑑\zeta \equiv {\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{h(\theta )}{\zeta -z}}𝑑\zeta \equiv 0,$

   where $h(\theta )\equiv f(\theta )-g(\theta )$. We shall prove that $h(\theta )\equiv 0$. Since we can write,

   ${\displaystyle \frac{1}{\zeta -z}}={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{z^n}{{\zeta }^{n+1}}},$

   then

   ${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{h(\theta )}{\zeta -z}}𝑑\zeta ={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{h(\theta )}{{\zeta }^{n+1}}}{z}^{n}d\zeta ={\displaystyle \frac{1}{2\pi i}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}(a_n-i{b}_n)z^n,$ (2)

   the complex form of Fourier series where the coefficients are given by (we are using the Euler’s formula)

   $a_n-i{b}_n={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{h(\theta )}{{\zeta }^{n+1}}}𝑑\zeta ={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }}h(\theta )e^{-in\theta }𝑑\theta .$

   But (2) vanishes for all values of $z$, therefore $a_n=b_n=0,n\in \mathbb{N}$ and a reference to Fourier complex expansion

   $h(\theta )={\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}}{c}_n{e}^{in\theta },c_n={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }}h(t)e^{-int}𝑑t,n\in \mathbb{Z},$

   shows that all Fourier coefficients of the function $h(\theta )$ vanish, and hence $h(\theta )\equiv 0$.

2. $|z|>1$. By analytic continuation, we have

   ${\displaystyle \frac{1}{\zeta -z}}={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}-{\displaystyle \frac{{\zeta }^n}{z^{n+1}}},$

   so that

   ${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{h(\theta )}{\zeta -z}}𝑑\zeta =-{\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{\zeta }^{n-1}h(\theta )}{z^n}}d\zeta =-{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{a_n+i{b}_n}{z^n}},$ (3)

   where

   $a_n+i{b}_n={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\zeta }^{n-1}h(\theta )𝑑\zeta ={\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }}h(\theta )e^{in\theta }𝑑\theta ,n\in \mathbb{N}\backslash \{0\}.$

   Since (3) vanishes for all values of $|z|>1$, $a_n=b_n=0,n\in \mathbb{N}\backslash \{0\}$. Thus, all Fourier coefficients of $h(\theta )$, with the possible exception of $a_0$, vanish and hence

   $g(\theta )=f(\theta )+const.$

∎

Moreover, from this theorem if $|z|>1$ and in addition to (1) we have the equality

${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{f(\theta )}{\zeta }}𝑑\zeta ={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{g(\theta )}{\zeta }}𝑑\zeta ,$

then $f(\theta )=g(\theta )$.

Corollary 1.

Given the continuous real functions $f_{\mathrm{1}}\mathrm{,}{f}_{\mathrm{2}}\mathrm{,}{g}_{\mathrm{1}}\mathrm{,}{g}_{\mathrm{2}}$ and the following simultaneous equalities for all values of $z$.

${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{f_1+i{f}_2}{\zeta -z}}𝑑\zeta ={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{g_1+i{g}_2}{\zeta -z}}𝑑\zeta ,$

${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{f_1-i{f}_2}{\zeta -z}}𝑑\zeta ={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_{\gamma }}{\displaystyle \frac{g_1-i{g}_2}{\zeta -z}}𝑑\zeta ,$

then

and

$g_1=f_1+const.,g_2=f_2+const.,if\mathit{\hspace{1em}}|z|>1.$

By adding and substracting those equalities, this corollary follows from Harnack theorem.

Title	Harnack theorem
Canonical name	HarnackTheorem
Date of creation	2013-03-22 16:02:55
Last modified on	2013-03-22 16:02:55
Owner	perucho (2192)
Last modified by	perucho (2192)
Numerical id	8
Author	perucho (2192)
Entry type	Theorem
Classification	msc 30D10