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Theodorsen's integral equation [[#References|[a7]]] is a well-known tool for computing numerically the [[Conformal mapping|conformal mapping]] $g$ of the unit disc $D$ onto a star-like region $\Delta$ given by the polar coordinates $\tau$, $\rho ( \tau )$ of its boundary $\Gamma$. The mapping $g$ is assumed to be normalized by $g ( 0 ) = 0$, $g ^ { \prime } ( 0 ) > 0$. It is uniquely determined by its boundary correspondence function $\theta$, which is implicitly defined by

\begin{equation*} g ( e ^ { i t } ) = \rho ( \theta ( t ) ) e ^ { i \theta ( t ) } ( \forall t \in \mathbf{R} ), \end{equation*}

\begin{equation*} \int _ { 0 } ^ { 2 \pi } \theta ( t ) d t = 2 \pi ^ { 2 }. \end{equation*}

Theodorsen's equation follows from the fact that the function $h ( w ) : = \operatorname { log } ( g ( w ) / w )$ is analytic in $D$ and can be extended to a [[Homeomorphism|homeomorphism]] of the closure $\overline{ D }$ onto the closure $\overline{\Delta}$. It simply states that the $2 \pi$-periodic function $y$: $t \mapsto \theta - t$ is the conjugate periodic function of $x$: $t \mapsto \operatorname { log } \rho ( \theta ( t ) )$, that is, $y = K x$, where $K$ is the conjugation operator defined on $L [ 0,2 \pi ]$ by the principal value integral

\begin{equation*} ( K x ) ( t ) : = \frac { 1 } { 2 \pi } \text{P} \cdot \text{V} \cdot \int _ { 0 } ^ { 2 \pi } x ( s ) \operatorname { cot } \frac { t - s } { 2 } d s \ (a.e.) . \end{equation*}

When restricted to $L _ { 2 } [ 0,2 \pi ]$, $K$ is a skew-symmetric endomorphism of norm $1$ with a very simple diagonal representation in Fourier space: when $x$ has the real Fourier coefficients $a_0 , a_1 , \dots$, $b _ { 1 } , b _ { 2 } , \dots$, then $y$ has the coefficients $0 , - b _ { 1 } , - b _ { 2 } , \dots$, $a _ { 1 } , a _ { 2 } , \dots$.

Hence, while Theodorsen's integral equation is normally written as

\begin{equation*} \theta ( t ) - t = \frac { 1 } { 2 \pi } \operatorname {P} \cdot \operatorname {V}\cdot \int _ { 0 } ^ { 2 \pi } \operatorname { log } \rho ( \theta ( s ) ) \operatorname { cot } \frac { t - s } { 2 } d s, \end{equation*}

for practical purposes the conjugation is executed by transformation to Fourier space: $x$ is approximated by a [[Trigonometric polynomial|trigonometric polynomial]] of degree $N$, whose Fourier coefficients are quickly found by the fast Fourier transform, which then can also be applied to determine values at $2 N$ equi-spaced points of the trigonometric polynomial that approximates $y = K x$ (cf. also [[Fourier series|Fourier series]]). Before the fast Fourier transform became the standard tool for this discrete conjugation process, the transition from the values of $x$ to those of $y$ was based on multiplication by a matrix, called the Wittich matrix in [[#References|[a1]]]. The fast Fourier transform meant a cost reduction from $O ( N ^ { 2 } )$ to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130070/t13007043.png"/> operations per iteration.

Until the end of the 1970s the recommendation was to solve a so-obtained discrete version of Theodorsen's equation by fixed-point (Picard) iteration, an approach that is limited to Jordan regions with piecewise differentiable boundary satisfying $| \rho ^ { \prime } / \rho | < 1$, and is very slow when the bound $1$ is nearly attained. Other regions, like those from airfoil design, which was the standard application targeted by T. Theodorsen, could be handled by using first a suitable preliminary conformal mapping, which turned the exterior of the wing cross-section into the exterior of a Jordan curve that is close to a circle; see [[#References|[a6]]], Chapt. 10. Moreover, for this application, the equation has to be modified slightly to map the exterior of the disc onto the exterior of a Jordan curve.

M. Gutknecht [[#References|[a4]]], [[#References|[a2]]] extended the applicability of Theodorsen's equation by applying more refined iterative methods and discretizations, and O. Hübner [[#References|[a5]]] improved the convergence order from linear to quadratic by adapting R. Wegmann's treatment of a similar equation obtained by choosing $h ( w ) : = g ( w ) / w$ instead. Wegmann's method [[#References|[a9]]], [[#References|[a10]]] applies the [[Newton method|Newton method]] and solves the linear equation for the corrections by interpreting it as a [[Riemann–Hilbert problem|Riemann–Hilbert problem]] that can be solved with four fast Fourier transforms.

A common framework for conformal mapping methods based on function conjugation is given in [[#References|[a3]]]; Theodorsen's restriction to regions given in polar coordinates can be lifted. Both Theodorsen's [[#References|[a8]]] and Wegmann's [[#References|[a11]]] equations and methods can be extended to the doubly connected case.

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">D. Gaier, "Konstruktive Methoden der konformen Abbildung" , Springer  (1964) {{ZBL|0132.36702}}</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top">M.H. Gutknecht, "Numerical experiments on solving Theodorsen's integral equation for conformal maps with the fast Fourier transform and various nonlinear iterative methods"  ''SIAM J. Sci. Statist. Comput.'' , '''4'''  (1983)  pp. 1–30</td></tr>
<tr><td valign="top">[a3]</td> <td valign="top">  M.H. Gutknecht, "Numerical conformal mapping methods based on function conjugation"  ''J. Comput. Appl. Math.'' , '''14'''  (1986)  pp. 31–77</td></tr>
<tr><td valign="top">[a4]</td> <td valign="top">  M.H. Gutknecht, "Solving Theodorsen's integral equation for conformal maps with the fast Fourier transform and various nonlinear iterative methods"  ''Numer. Math.'' , '''36'''  (1981)  pp. 405–429</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  O. Hübner,   "The Newton method for solving the Theodorsen equation"  ''J. Comput. Appl. Math.'' , '''14'''  (1986)  pp. 19–30</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  P.K. Kythe,   "Computational conformal mapping" , Birkhäuser  (1998)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  T. Theodorsen,   "Theory of wing sections of arbitrary shape"  ''Rept. NACA'' , '''411'''  (1931)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  T. Theodorsen,   I.E. Garrick,   "General potential theory of arbitrary wing sections"  ''Rept. NACA'' , '''452'''  (1933)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  R. Wegmann,   "Ein Iterationsverfahren zur konformen Abbildung"  ''Numer. Math.'' , '''30'''  (1978)  pp. 453–466</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  R. Wegmann,   "An iterative method for conformal mapping"  ''J. Comput. Appl. Math.'' , '''14'''  (1986)  pp. 7–18  (Translated from German)  (English translation of [9])</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  R. Wegmann,   "An iterative method for the conformal mapping of doubly connected regions"  ''J. Comput. Appl. Math.'' , '''14'''  (1986)  pp. 79–98</td></tr></table>