# Conformal mapping

A continuous mapping preserving the form of infinitesimal figures.

## Fundamental concepts.

A continuous mapping $w = f ( z)$ of a domain $G$ in an $n$- dimensional Euclidean space $( n \geq 2)$ into the $n$- dimensional Euclidean space is called conformal at a point $z _ {0} \in G$ if it has the properties of constancy of dilation and preservation of angles at this point. The property of constancy of dilation at $w = f ( z)$ means that the ratio $| f ( z) - f ( z) | / | z - z _ {0} |$ of the distance between the images $f ( z)$ and $f ( z _ {0} )$ of the points $z$ and $z _ {0}$ to the distance between $z$ and $z _ {0}$ tends to a definite limit $k = k ( z _ {0} , f )$ as $z$ tends to $z _ {0}$ in an arbitrary way. The number $k$ is called the coefficient of dilation at $z _ {0}$ for the given mapping. The property of preservation (conservation) of angles at $z _ {0} \in G$ by $w = f ( z)$ means that any pair of continuous curves $l _ {1} , l _ {2}$ in $G$ intersecting at $z _ {0}$ at an angle $\alpha$( that is, their tangents at $z _ {0}$ form an angle $\alpha$) is taken under the given mapping to a pair of continuous curves $L _ {1} , L _ {2}$ intersecting at the same angle $\alpha$ at $w _ {0} = f ( z _ {0} )$. A continuous mapping of a domain $G$ is called conformal if it is conformal at each point of this domain. By definition, a conformal mapping of a domain $G$ is required to be continuous and conformal only at the interior points of $G$; if one speaks about a conformal mapping of a closed domain, then, as a rule, one has in mind a continuous mapping of the closed domain that is conformal at interior points.

In the most important case $n = 2$, the domain $G$ and its image $f ( G)$ under the mapping $f$ lie in a plane, which is conveniently regarded as the complex $z$- plane $\mathbf C$; accordingly, $w = f ( z)$ is a complex-valued function of the complex variable $z \in G$. Furthermore, if $w = f ( z)$ preserves angles at a point $z _ {0}$, then the curvilinear angles with vertex $z _ {0}$ either retain both their size and sign under this mapping or retain their size and change their sign. In the first case one says that the mapping is conformal of the first kind at $z _ {0}$, and in the second case — conformal of the second kind. If a function $w = f ( z)$ defines a conformal mapping of the second kind at a point $z _ {0}$, then the complex-conjugate mapping $w = \overline{ {f ( z) }}\;$ is conformal of the first kind at $z _ {0}$, and conversely. Therefore, only conformal mappings of the first kind are studied, and these are meant when one speaks of conformal mappings without specifying their kind.

If a mapping $w = f ( z)$ is conformal at $z _ {0}$, then as $z \rightarrow z _ {0}$, the ratio $( f ( z) - f ( z _ {0} ))/( z - z _ {0} )$ tends to a finite limit, that is, the derivative $f ^ { \prime } ( z _ {0} )$ exists. Under the additional assumption that $f ^ { \prime } ( z _ {0} ) \neq 0$, the converse is also true.

Thus, if $f ^ { \prime } ( z _ {0} )$ exists and is non-zero, then under $w = f ( z)$ each infinitesimal vector with origin at $z _ {0}$ is dilated $k ( z _ {0} , f ) = | f ^ { \prime } ( z _ {0} ) |$ times, is rotated through an angle $\mathop{\rm arg} f ^ { \prime } ( z _ {0} )$ and is shifted by the vector $f ( z _ {0} ) - z _ {0}$; infinitesimal discs centred at $z _ {0}$ are taken to infinitesimal discs.

A mapping $w = f ( z)$ is conformal in a domain $G$ of the complex plane $\mathbf C$ if and only if the function $f ( z)$, $z \in G$, is analytic and $f ^ { \prime } ( z) \neq 0$ in $G$. In order that a mapping $w = f ( z)$ be conformal (or that $f ( z)$ be analytic) in a domain $G$, it suffices that $f ( z)$ be continuous and that at each point $z \in G$ it has the property of preservation of angles (the property of preservation of signs as well as sizes of angles). If, instead, one requires that a continuous mapping $w = f ( z)$, $z \in G$, be univalent (that is, one-to-one) and possess constancy of dilation at every point, then this mapping is conformal of the first or second kind, so that either $f ( z)$ or $\overline{ {f ( z) }}\;$ is an analytic function with non-zero derivative throughout $G$. For the case when $f ( z)$ is analytic in some neighbourhood of a point $z _ {0} \in \mathbf C$, the following three properties are equivalent: a) $w = f ( z)$ is a conformal mapping (of the first kind) at $z _ {0}$; b) $f ( z)$ is (locally) a univalent function at $z _ {0}$; or c) $f ^ { \prime } ( z _ {0} ) \neq 0$. Every univalent analytic function in a domain $G$ conformally maps $G$ onto a domain $f ( G)$ of the same connectivity; furthermore, the inverse function $f ^ { - 1 } ( z)$ is a univalent analytic function in $f ( G)$ and has a non-zero derivative. There also exist non-univalent conformal mappings (for example, $w = z ^ {4}$ is a non-univalent conformal mapping in the half-plane $\mathop{\rm Im} z > 0$; and $w = e ^ {z}$ is a non-univalent analytic mapping in the whole of $\mathbf C$).

In the theory and application of conformal mappings in the plane the principal question is that of the possibility of mapping a given domain onto another by a univalent conformal mapping, and in practical applications, the question of the possibility of achieving this using relatively simple functions. The first problem is solved affirmatively for the case of simply-connected domains with non-empty boundaries that do not degenerate into points by the Riemann mapping theorem (cf. Riemann theorem). The second problem is solved for some domains of special type by applying elementary functions of a complex variable (see below), the Christoffel–Schwarz formula for mapping a half-plane or a disc onto a polygon, and applications of the reflection principle and approximation methods for conformal mappings. According to the Riemann mapping theorem, all simply-connected domains in the extended complex plane with non-empty boundaries that do not degenerate into points are conformally equivalent. For conformal mappings of multiply-connected domains the situation is different. Since a univalent conformal mapping of some domain $G _ {1}$ onto another $G _ {2}$ is one-to-one, continuous and has a continuous inverse, in order that such a mapping exists it is necessary that $G _ {1}$ and $G _ {2}$ have the same order of connectivity, that is, they must both be either simply connected, or doubly connected, etc., or infinitely connected. However, this necessary condition is not sufficient, as already becomes apparent in the case of simply-connected domains. Thus, the disc $| z | < 1$ cannot be univalently and conformally mapped onto the finite plane $\mathbf C$, while the extended complex plane $\overline{\mathbf C}\;$ cannot be conformally and univalently mapped onto the disc $| z | < 1$ or the plane $\mathbf C$( in fact, not even a topological mapping can be found in the last two cases). The situation is even more stringent in the case of multiply-connected domains. For example, an annulus $G _ {1} = \{ {z } : {r _ {1} < | z | < R _ {1} } \}$ can be univalently and conformally mapped onto another annulus $G _ {2} = \{ {z } : {r _ {2} < | z | < R _ {2} } \}$, $r _ {1} > 0$, $r _ {2} > 0$, if and only if these annuli are similar, that is, if $R _ {1} / r _ {1} = R _ {2} / r _ {2}$; in this case every conformal mapping of $G _ {1}$ onto $G _ {2}$ is an entire linear function of the form $e ^ {i \beta } r _ {2} z/r _ {1}$, where $\beta$ is a real number. However, every finitely-connected domain $G$ with a non-empty boundary in the extended complex plane can be univalently and conformally mapped onto one of the so-called canonical domains of the same connectivity containing the point $\infty$, namely: onto the extended complex plane with finitely many horizontal slits; onto the extended complex plane with the exception of finitely many non-intersecting closed discs; or onto the extended complex plane with the exception of closed arcs of logarithmic spirals of given slope $\theta$. Here the individual slits, discs and arcs of spirals may degenerate into points. If it is required that a given point $a \in G$ is taken under such a mapping to $\infty$ and that, as $z \rightarrow a$, the relation $f ( z) = ( z - a) ^ {-} 1 + O ( z - a)$ holds, for $a \neq \infty$, or the relation $f ( z) = z + O ( 1/z)$ holds, for $a = \infty$, then in the case of the first two canonical domains, the mapping function $w = f ( z)$ exists, is defined uniquely and is called the canonical conformal mapping. Similar theorems also hold for infinitely-connected domains.

Every univalent mapping of a domain $G _ {1}$ bounded by a finite number of non-intersecting circles (and here a straight line is considered to be a circle of infinite radius) onto a domain $G _ {2}$ of the same type is a fractional-linear mapping. In the theory of analytic functions, non-univalent mappings by analytic functions between domains of different connectivities are also considered. These include: a conformal mapping of a disc onto a multiply-connected domain; a mapping of an $n$- connected domain onto an $n$- sheeted disc; and, more generally, a mapping from one Riemann surface onto another.

In the theory and application of conformal mappings an important role is played by so-called normalization conditions, or uniqueness conditions, for conformal mappings. These enable one to select a unique function from the infinite class of conformal mappings under consideration of one given domain onto another (in the case of simply-connected domains) or of a given domain onto a canonical domain of specified type (in the case of arbitrarily-connected domains). The most commonly used normalization conditions for conformal mappings in the case of simply-connected domains $G _ {1}$, $G _ {2}$ with non-empty boundaries $\Gamma _ {1}$, $\Gamma _ {2}$ that do not degenerate into points, respectively, are: 1) a given finite point $a \in G _ {1}$ is taken to a given finite point $b \in G _ {2}$, where $\mathop{\rm arg} f ^ { \prime } ( a) = \alpha$, $\alpha$ being a pre-assigned real number, $0 \leq \alpha < 2 \pi$( see Riemann theorem on conformal mapping); 2) a given point $a \in G _ {1}$ is taken to a given point $b \in G _ {2}$ and a given accessible boundary point (prime end, boundary element or limit element; cf. Limit elements; Attainable boundary point) $\zeta _ {1}$ of $G _ {1}$ is taken to a given accessible boundary point (prime end) $\omega _ {1}$ of $G _ {2}$( see Conformal mapping, boundary properties of a); or 3) three given distinct accessible boundary points (prime ends) $\zeta _ {1} , \zeta _ {2} , \zeta _ {3}$ of a domain $G _ {1}$ are taken, respectively, to given accessible boundary points (prime ends) $\omega _ {1} , \omega _ {2} , \omega _ {3}$ of the domain $G _ {2}$, where, if on going along $\Gamma _ {1}$ from $\zeta _ {1}$ to $\zeta _ {3}$ via $\zeta _ {2}$ the domain $G _ {1}$ is on the left (or right), then on going along $\Gamma _ {2}$ from $\omega _ {1}$ to $\omega _ {3}$ via $\omega _ {2}$, the domain $G _ {2}$ must also be on the left (on the right). The last two types of normalization are most often applied in the case of domains bounded by closed Jordan curves, since in this case the notions of accessible boundary points and prime ends of the domain are equivalent at boundary points of the domain. Normalization conditions in the case of a mapping of an arbitrarily-connected domain onto a canonical domain have already been discussed above.

The conformal mappings of domains in an $n$- dimensional Euclidean space for $n \geq 3$ form the very narrow class of so-called Möbius mappings, each of which is either a linear similarity mapping or a composite of such a linear similarity mapping and an inversion (that is, a symmetry with respect to some sphere in the space, or a mapping of inverse radii) (Liouville's theorem). A significantly larger class of mappings for $n \geq 2$ is formed by the so-called quasi-conformal mappings (cf. Quasi-conformal mapping). Under these mappings the forms of infinitesimal figures are distorted, but within bounded limits, in particular, the sizes of angles are changed within bounded limits, while infinitesimal balls are taken into infinitesimal ellipsoids with bounded ratio of the major to the minor axis.

A large part is played by conformal mappings of two-dimensional domains not only on planar surfaces but also for domains lying on smooth surfaces. Examples of such conformal mappings are given by stereographic projection and Mercator projection of a sphere onto the plane. They were discovered and applied in cartography (see Cartography, mathematical problems in; Cartographic projection). It should be noted that at that time posing the problem of conformally mapping surfaces led, in its general form, to the origin and development of the general theory of surfaces. Conformal mappings find wide application in the theory of functions, potential theory, in the solution of boundary value problems for the equations of mathematical physics, and above all in the solution of the first boundary value problem for the Laplace and Poisson equations.

## Conformal mappings of certain simply-connected domains.

Dilations, rotations and parallel translations of domains in the complex plane are realized by entire linear functions of the form $w = az + b$. Univalent conformal mappings of half-planes, discs and exteriors of discs onto one another are realized by fractional-linear transformations. Here, given any three different points $\zeta _ {1} , \zeta _ {2} , \zeta _ {3}$ on the boundary of one of the domains $G _ {1}$( the enumeration is such that on going around the boundary of $G _ {1}$ from $\zeta _ {1}$ to $\zeta _ {3}$ via $\zeta _ {2}$, $G _ {1}$ is on the left) and three different points $\omega _ {1} , \omega _ {2} , \omega _ {3}$ on the boundary of another such domain $G _ {2}$( with a similar enumeration), there exists a unique fractional-linear transformation $w = L ( z)$ univalently and conformally mapping $G _ {1}$ onto $G _ {2}$ with normalization conditions of the third type: $L ( \zeta _ {k} ) = \omega _ {k}$, $k = 1, 2, 3$. This function can be found from the equation

$$\frac{w - \omega _ {1} }{w - \omega _ {2} } : \ \frac{\omega _ {3} - \omega _ {1} }{\omega _ {3} - \omega _ {2} } = \ \frac{z - \zeta _ {1} }{z - \zeta _ {2} } : \ \frac{\zeta _ {3} - \zeta _ {1} }{\zeta _ {3} - \zeta _ {2} } ,$$

in which each numerator or denominator must be replaced by the number 1 if the point $\omega _ {k} = \infty$ or the point $\zeta _ {k} = \infty$ enters in its description. In particular, the general form of mappings of the unit disc $D = \{ {z } : {| z | < 1 } \}$ onto itself is

$$w = L _ {1} ( z) = \ e ^ {i \alpha } \frac{z - a }{1 - \overline{a}\; z } ,$$

while the general form of a mapping from the upper half-plane $P = \{ {z } : { \mathop{\rm Im} z > 0 } \}$ onto this disc is:

$$w = L _ {2} ( z) = \ e ^ {i \alpha } \frac{z - c }{z - \overline{c}\; } .$$

Here $a$ and $c$ are, respectively, the pre-images of 0 under these mappings, and $\alpha = \mathop{\rm arg} L _ {1} ^ \prime ( a) = \mathop{\rm arg} L _ {2} ^ \prime ( c)$, $0 \leq \alpha < 2 \pi$. The numbers $a$, $| a | < 1$, $c$, $\mathop{\rm Im} c > 0$, and $\alpha$, $0 \leq \alpha < 2 \pi$, can be arbitrarily prescribed. Thus, the above general form of univalent conformal mappings of the unit disc and upper half-plane onto the unit disc enables one to take into account the normalization conditions of the first type in a simple way. Normalizations of the second type with $b = 0$ are also easily fulfilled if one uses the above general form with the given $a$( or $c$), after which it only remains to choose the factor $e ^ {i \alpha }$ from the correspondence condition of the given boundary points $\zeta$ and $\omega$.

The simplicity of fulfilling the normalization conditions under a mapping of the unit disc onto itself or of the upper half-plane onto itself lies at the basis of the following widely used device, by means of which normalization conditions are taken into account for univalent conformal mappings between arbitrary domains $G _ {1}$, $G _ {2}$ with non-empty non-degenerate boundaries. Namely, both domains $G _ {1}$ and $G _ {2}$ are, somehow, conformally and univalently mapped onto $D$( or onto $P$) by means of certain functions $w = f _ {1} ( z)$ and $w = f _ {2} ( z)$, respectively, after which the problem of mapping $G _ {1}$ onto $G _ {2}$ with certain normalization conditions reduces to that of finding a fractional-linear transformation $w = L ( z)$ of $D$( or of $P$) onto itself with the corresponding normalization conditions being fulfilled. If the function $L$ has been found, then $f ( z) = f _ {2} ^ { - 1 } ( L ( f _ {1} ( z)))$ solves the original problem. In view of this, only univalent conformal mappings of various domains onto the unit disc $D$, or onto the upper half-plane $P$, without any normalization conditions will be mentioned below.

1) The horizontal strip $\{ {z = x + iy } : {0 < y < \pi } \}$ is mapped onto the upper half-plane by the function $w = e ^ {z} = e ^ {x} ( \cos y + i \sin y)$. This function takes a horizontal line $y = c$ to the ray $\mathop{\rm Arg} w = c$, and a vertical segment $\{ {x + iy } : {x = d, \alpha \leq y \leq \beta } \}$ onto the circular arc $\{ {w } : {| w | = e ^ {d\ } \alpha \leq \mathop{\rm Arg} w \leq \beta } \}$.

2) The vertical strip $\{ {x + iy } : {- \pi /4 < x < \pi /4 } \}$ is mapped by the function $w = \mathop{\rm tan} z$ onto the disc $D$. Here a vertical line $x = c$, $- \pi /4 \leq c \leq \pi /4$, is taken to the "meridian" arc of the circle with end points $i$ and $- i$, passing through the point $w = \mathop{\rm tan} c$, while a horizontal segment $\{ {x + iy } : {- \pi /4 \leq x \leq \pi /4, y = d } \}$, $- \infty < d < \infty$, is taken to the "latitudinal" arc of the circle orthogonal to the "meridian" joining the left half of the unit circle to the right half and passing through the point $z = i \mathop{\rm tanh} d$.

3) The half-strip $\{ {x + iy } : {- \pi /2 \langle x < \pi /2, y \rangle 0 } \}$ is mapped onto $P$ by the function $y = \sin z$. Here, horizontal segments are taken into arcs of ellipses with foci $- 1, + 1$, while vertical segments are taken into arcs of hyperbolas with the same foci.

4) The sector $V _ \alpha = \{ {z } : {0 < \mathop{\rm arg} z < \alpha } \}$, $0 < \alpha \leq 2 \pi$, is mapped onto the sector $V _ \alpha = \{ {z } : {0 < \mathop{\rm arg} z < \beta } \}$, $0 < \beta \leq 2 \pi$( $V _ \pi = P$), by the function

$$w = \mathop{\rm exp} \left \{ { \frac \beta \alpha } \mathop{\rm ln} z \right \} =$$

$$= \ | z | ^ {\beta / \alpha } \left ( \cos \left ( { \frac \beta \alpha } \mathop{\rm arg} z \right ) + i \ \sin \left ( { \frac \beta \alpha } \mathop{\rm arg} z \right ) \right ) ,$$

i.e. by a single-valued analytic branch of the function $w = z ^ {\beta / \alpha }$. Here a ray $\mathop{\rm arg} z = c$ is taken to the ray $\mathop{\rm arg} w = {\beta c } / \alpha$, and an arc of the circle $| z | = d$ to an arc of the circle $| w | = d ^ {\beta / \alpha }$.

5) The interior or exterior of the digon with vertices $a$ and $b$( $a \neq b$) formed by two arcs of circles, or an arc of a circle and a straight line segment, having the same end points as these points can be mapped onto $P$ in the following way. First the given domain is mapped onto the sector $V$ with vertex at $O$ by the fractional-linear mapping $w _ {1} = ( z - a)/( z - b)$, then $V$ is taken onto some sector $V _ \alpha$( see 4) above) by the rotation $w _ {2} = e ^ {i \gamma } w _ {1}$ over some angle $\gamma$, after which one obtains the transformation in 4) with $\beta = \pi$.

6) The exterior of the ellipse $( {x ^ {2} } / {a ^ {2} } ) + ( {y ^ {2} } / {b ^ {2} } ) = 1$ with foci at distance $c = \sqrt {a ^ {2} - b ^ {2} } > 0$ is mapped by a single-valued branch of the analytic function (see Zhukovskii function)

$$w = { \frac{1}{c} } ( z + \sqrt {z ^ {2} - c ^ {2} } ),$$

chosen subject to the condition $| w | < 1$, onto the disc $| w | < c/( a + b)$, and by the other branch, chosen subject to the condition $| w | > 1$, onto the domain $| w | > ( a + b)/c$. These same branches map the extended plane $\overline{\mathbf C}\;$ with slit at $[- c, c]$, respectively, onto the interior and exterior of the unit circle $| z | = 1$. The interior of an ellipse cannot be mapped onto $D$ or $P$ by a composite of elementary functions; this mapping can be realized by the composite of elementary functions and the elliptic sine $\mathop{\rm sn} z$.

7) The part $G$ of $\mathbf C$ between the branches of the hyperbola $( x ^ {2} /a ^ {2} ) - ( y ^ {2} /b ^ {2} ) = 1$ with foci at a distance $c = \sqrt {a ^ {2} + b ^ {2} }$ is mapped onto $P$ by the function

$$w = \left ( e ^ {- i \gamma } { \frac{1}{c} } ( z + \sqrt {z ^ {2} - c ^ {2} } ) \right ) ^ {\pi /( \pi - 2 \gamma ) } ,$$

where the single-valued analytic branch of the function $t = z + \sqrt {z ^ {2} - c ^ {2} }$ in $G$ is selected by the condition $\mathop{\rm Im} t > 0$, $\gamma = \mathop{\rm arctan} ( b/a)$,

$$\zeta ^ {\pi /( \pi - 2 \gamma ) } =$$

$$= \ | \zeta | ^ {\pi /( \pi - 2 \gamma ) } \left ( \cos \left ( { \frac \pi {\pi - 2 \gamma } } \mathop{\rm arg} \zeta \right ) + i \sin \left ( { \frac \pi {\pi - 2 \gamma } } \mathop{\rm arg} \zeta \right ) \right ) .$$

The interior of the right sheet of this hyperbola is mapped onto the upper half-plane by a single-valued branch of the analytic function

$$i \cosh \ \left ( { \frac \pi {2 \gamma } } \ \cosh ^ {-} 1 \ { \frac{z}{c} } \right ) ,$$

where $\cosh ^ {-} 1 \zeta$ denotes the (unique) solution of the equation $\cosh \omega = \zeta$ belonging to the strip $\{ {z } : {0 < \mathop{\rm Im} z < \pi } \}$ with the positive real semi-axis adjoined.

8) The exterior of the parabola $y ^ {2} = 2px$ is mapped onto $P$ by a single-valued analytic branch of the function

$$w = \ \sqrt {z - { \frac{p}{2} } } - i \sqrt { \frac{p}{2} } ,$$

chosen subject to the condition $\mathop{\rm Im} w > 0$, that is, by the function

$$w = \sqrt { \left | z - { \frac{p}{2} } \right | } \ \left \{ \cos { \frac{1}{2} } \ \mathop{\rm arg} \left ( z - { \frac{p}{2} } \right ) \right . +$$

$$+ \left . i \sin { \frac{1}{2} } \mathop{\rm arg} \left ( z - { \frac{p}{2} } \right ) \right \} - i \sqrt { \frac{p}{2} } ,$$

where the value of the square root is taken to be positive. The interior of this parabola is mapped onto $P$ by the single-valued analytic function

$$w = i \cosh \pi \sqrt { { \frac{z}{2p} } - { \frac{1}{4} } } ,$$

where

$$\sqrt \zeta = \ \sqrt {| \zeta | } \left ( \cos \ { \frac{1}{2} } \ \mathop{\rm arg} \zeta + i \sin \ { \frac{1}{2} } \ \mathop{\rm arg} \zeta \right ) .$$

9) The rectangle $Q = \{ {x + iy } : {- a < x < a, 0 < y < b } \}$ is mapped onto $P$ by the elliptic sine, and $P$ is mapped onto $Q$ by a single-valued analytic branch of the function

$$w = c \int\limits _ { 0 } ^ { z } \frac{ds }{\sqrt {( 1 - \zeta ^ {2} ) ( 1 - k ^ {2} \zeta ^ {2} ) } } ,$$

where $k$ depends on the ratio $a/b$, and $c$ on the quantity $a/K$, where

$$K = \int\limits _ { 0 } ^ { 1 } \frac{dt }{\sqrt {( 1 - t ^ {2} ) ( 1 - k ^ {2} t ^ {2} ) } } .$$

How to Cite This Entry:
Conformal mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_mapping&oldid=46455
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article