View source for Conformal mapping

A continuous mapping preserving the form of infinitesimal figures.

==Fundamental concepts.==
A continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c0247801.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c0247802.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c0247803.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c0247804.png" /> into the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c0247805.png" />-dimensional Euclidean space is called conformal at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c0247806.png" /> if it has the properties of constancy of dilation and preservation of angles at this point. The property of constancy of dilation at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c0247807.png" /> means that the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c0247808.png" /> of the distance between the images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c0247809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478010.png" /> of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478012.png" /> to the distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478014.png" /> tends to a definite limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478015.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478016.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478017.png" /> in an arbitrary way. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478018.png" /> is called the coefficient of dilation at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478019.png" /> for the given mapping. The property of preservation (conservation) of angles at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478020.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478021.png" /> means that any pair of continuous curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478023.png" /> intersecting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478024.png" /> at an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478025.png" /> (that is, their tangents at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478026.png" /> form an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478027.png" />) is taken under the given mapping to a pair of continuous curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478028.png" /> intersecting at the same angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478030.png" />. A continuous mapping of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478031.png" /> is called conformal if it is conformal at each point of this domain. By definition, a conformal mapping of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478032.png" /> is required to be continuous and conformal only at the interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478033.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478034.png" />, the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478035.png" /> and its image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478036.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478037.png" /> lie in a plane, which is conveniently regarded as the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478038.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478039.png" />; accordingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478040.png" /> is a complex-valued function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478041.png" />. Furthermore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478042.png" /> preserves angles at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478043.png" />, then the curvilinear angles with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478044.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478045.png" />, and in the second case — conformal of the second kind. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478046.png" /> defines a conformal mapping of the second kind at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478047.png" />, then the complex-conjugate mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478048.png" /> is conformal of the first kind at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478049.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478050.png" /> is conformal at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478051.png" />, then as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478052.png" />, the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478053.png" /> tends to a finite limit, that is, the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478054.png" /> exists. Under the additional assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478055.png" />, the converse is also true.

Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478056.png" /> exists and is non-zero, then under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478057.png" /> each infinitesimal vector with origin at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478058.png" /> is dilated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478059.png" /> times, is rotated through an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478060.png" /> and is shifted by the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478061.png" />; infinitesimal discs centred at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478062.png" /> are taken to infinitesimal discs.

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478063.png" /> is conformal in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478064.png" /> of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478065.png" /> if and only if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478067.png" />, is analytic and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478068.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478069.png" />. In order that a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478070.png" /> be conformal (or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478071.png" /> be analytic) in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478072.png" />, it suffices that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478073.png" /> be continuous and that at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478074.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478076.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478077.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478078.png" /> is an analytic function with non-zero derivative throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478079.png" />. For the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478080.png" /> is analytic in some neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478081.png" />, the following three properties are equivalent: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478082.png" /> is a conformal mapping (of the first kind) at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478083.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478084.png" /> is (locally) a univalent function at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478085.png" />; or c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478086.png" />. Every univalent analytic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478087.png" /> conformally maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478088.png" /> onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478089.png" /> of the same connectivity; furthermore, the inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478090.png" /> is a univalent analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478091.png" /> and has a non-zero derivative. There also exist non-univalent conformal mappings (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478092.png" /> is a non-univalent conformal mapping in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478093.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478094.png" /> is a non-univalent analytic mapping in the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478095.png" />).

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|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|Christoffel–Schwarz formula]] for mapping a half-plane or a disc onto a polygon, and applications of the [[Reflection principle|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478096.png" /> onto another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478097.png" /> is one-to-one, continuous and has a continuous inverse, in order that such a mapping exists it is necessary that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c02478099.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780100.png" /> cannot be univalently and conformally mapped onto the finite plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780101.png" />, while the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780102.png" /> cannot be conformally and univalently mapped onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780103.png" /> or the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780104.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780105.png" /> can be univalently and conformally mapped onto another annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780108.png" />, if and only if these annuli are similar, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780109.png" />; in this case every conformal mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780110.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780111.png" /> is an entire linear function of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780112.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780113.png" /> is a real number. However, every finitely-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780114.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780115.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780116.png" />. Here the individual slits, discs and arcs of spirals may degenerate into points. If it is required that a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780117.png" /> is taken under such a mapping to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780118.png" /> and that, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780119.png" />, the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780120.png" /> holds, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780121.png" />, or the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780122.png" /> holds, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780123.png" />, then in the case of the first two canonical domains, the mapping function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780124.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780125.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780126.png" /> of the same type is a [[Fractional-linear mapping|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780127.png" />-connected domain onto an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780128.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780130.png" /> with non-empty boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780132.png" /> that do not degenerate into points, respectively, are: 1) a given finite point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780133.png" /> is taken to a given finite point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780134.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780136.png" /> being a pre-assigned real number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780137.png" /> (see [[Riemann theorem|Riemann theorem]] on conformal mapping); 2) a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780138.png" /> is taken to a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780139.png" /> and a given accessible boundary point (prime end, boundary element or limit element; cf. [[Limit elements|Limit elements]]; [[Attainable boundary point|Attainable boundary point]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780140.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780141.png" /> is taken to a given accessible boundary point (prime end) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780142.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780143.png" /> (see [[Conformal mapping, boundary properties of a|Conformal mapping, boundary properties of a]]); or 3) three given distinct accessible boundary points (prime ends) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780144.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780145.png" /> are taken, respectively, to given accessible boundary points (prime ends) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780146.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780147.png" />, where, if on going along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780148.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780149.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780150.png" /> via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780151.png" /> the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780152.png" /> is on the left (or right), then on going along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780153.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780154.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780155.png" /> via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780156.png" />, the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780157.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780158.png" />-dimensional Euclidean space for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780159.png" /> 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780160.png" /> is formed by the so-called quasi-conformal mappings (cf. [[Quasi-conformal mapping|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|stereographic projection]] and Mercator projection of a sphere onto the plane. They were discovered and applied in cartography (see [[Cartography, mathematical problems in|Cartography, mathematical problems in]]; [[Cartographic projection|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780161.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780162.png" /> on the boundary of one of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780163.png" /> (the enumeration is such that on going around the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780164.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780165.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780166.png" /> via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780167.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780168.png" /> is on the left) and three different points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780169.png" /> on the boundary of another such domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780170.png" /> (with a similar enumeration), there exists a unique fractional-linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780171.png" /> univalently and conformally mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780172.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780173.png" /> with normalization conditions of the third type: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780174.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780175.png" />. This function can be found from the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780176.png" /></td> </tr></table>

in which each numerator or denominator must be replaced by the number 1 if the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780177.png" /> or the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780178.png" /> enters in its description. In particular, the general form of mappings of the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780179.png" /> onto itself is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780180.png" /></td> </tr></table>

while the general form of a mapping from the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780181.png" /> onto this disc is:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780182.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780183.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780184.png" /> are, respectively, the pre-images of 0 under these mappings, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780185.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780186.png" />. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780187.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780188.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780189.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780190.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780191.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780192.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780193.png" /> are also easily fulfilled if one uses the above general form with the given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780194.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780195.png" />), after which it only remains to choose the factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780196.png" /> from the correspondence condition of the given boundary points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780197.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780198.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780199.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780200.png" /> with non-empty non-degenerate boundaries. Namely, both domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780202.png" /> are, somehow, conformally and univalently mapped onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780203.png" /> (or onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780204.png" />) by means of certain functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780206.png" />, respectively, after which the problem of mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780207.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780208.png" /> with certain normalization conditions reduces to that of finding a fractional-linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780209.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780210.png" /> (or of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780211.png" />) onto itself with the corresponding normalization conditions being fulfilled. If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780212.png" /> has been found, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780213.png" /> solves the original problem. In view of this, only univalent conformal mappings of various domains onto the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780214.png" />, or onto the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780215.png" />, without any normalization conditions will be mentioned below.

1) The horizontal strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780216.png" /> is mapped onto the upper half-plane by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780217.png" />. This function takes a horizontal line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780218.png" /> to the ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780219.png" />, and a vertical segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780220.png" /> onto the circular arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780221.png" />.

2) The vertical strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780222.png" /> is mapped by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780223.png" /> onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780224.png" />. Here a vertical line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780225.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780226.png" />, is taken to the  "meridian"  arc of the circle with end points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780227.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780228.png" />, passing through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780229.png" />, while a horizontal segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780230.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780231.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780232.png" />.

3) The half-strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780233.png" /> is mapped onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780234.png" /> by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780235.png" />. Here, horizontal segments are taken into arcs of ellipses with foci <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780236.png" />, while vertical segments are taken into arcs of hyperbolas with the same foci.

4) The sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780237.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780238.png" />, is mapped onto the sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780239.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780240.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780241.png" />), by the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780242.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780243.png" /></td> </tr></table>

i.e. by a single-valued analytic branch of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780244.png" />. Here a ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780245.png" /> is taken to the ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780246.png" />, and an arc of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780247.png" /> to an arc of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780248.png" />.

5) The interior or exterior of the digon with vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780249.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780250.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780251.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780252.png" /> in the following way. First the given domain is mapped onto the sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780253.png" /> with vertex at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780254.png" /> by the fractional-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780255.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780256.png" /> is taken onto some sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780257.png" /> (see 4) above) by the rotation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780258.png" /> over some angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780259.png" />, after which one obtains the transformation in 4) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780260.png" />.

6) The exterior of the ellipse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780261.png" /> with foci at distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780262.png" /> is mapped by a single-valued branch of the analytic function (see [[Zhukovskii function|Zhukovskii function]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780263.png" /></td> </tr></table>

chosen subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780264.png" />, onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780265.png" />, and by the other branch, chosen subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780266.png" />, onto the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780267.png" />. These same branches map the extended plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780268.png" /> with slit at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780269.png" />, respectively, onto the interior and exterior of the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780270.png" />. The interior of an ellipse cannot be mapped onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780271.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780272.png" /> by a composite of elementary functions; this mapping can be realized by the composite of elementary functions and the elliptic sine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780273.png" />.

7) The part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780274.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780275.png" /> between the branches of the hyperbola <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780276.png" /> with foci at a distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780277.png" /> is mapped onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780278.png" /> by the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780279.png" /></td> </tr></table>

where the single-valued analytic branch of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780280.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780281.png" /> is selected by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780282.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780283.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780284.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780285.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780286.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780287.png" /> denotes the (unique) solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780288.png" /> belonging to the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780289.png" /> with the positive real semi-axis adjoined.

8) The exterior of the parabola <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780290.png" /> is mapped onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780291.png" /> by a single-valued analytic branch of the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780292.png" /></td> </tr></table>

chosen subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780293.png" />, that is, by the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780294.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780295.png" /></td> </tr></table>

where the value of the square root is taken to be positive. The interior of this parabola is mapped onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780296.png" /> by the single-valued analytic function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780297.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780298.png" /></td> </tr></table>

9) The rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780299.png" /> is mapped onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780300.png" /> by the elliptic sine, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780301.png" /> is mapped onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780302.png" /> by a single-valued analytic branch of the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780303.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780304.png" /> depends on the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780305.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780306.png" /> on the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780307.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024780/c024780308.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,   B.V. Shabat,   "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.V. Keldysh,   "Conformal mappings of multiply-connected domains onto canonical domains"  ''Uspekhi Mat. Nauk'' , '''6'''  (1939)  pp. 90–119  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,   "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Lavrent'ev,   "Conformal mapping with applications to certain questions of mechanics" , Moscow-Leningrad  (1946)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V. Koppenfels,   "Praxis der konformen Abbildung" , Springer  (1959)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.I. Lavrik,   V.N. Savenkov,   "Handbook on conformal mapping" , Kiev  (1970)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.P. Fil'chakova,   "Conformal mapping of domains of special type" , Kiev  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  C. Carathéodory,   "Conformal representation" , Cambridge Univ. Press  (1932)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.V. Bitsadze,   "Fundamentals of the theory of analytic functions of a complex variable" , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  M.A. Lavrent'ev,   B.V. Shabat,   "Problems in hydrodynamics and their mathematical models" , Moscow  (1973)  (In Russian)</TD></TR></table>



====Comments====
For the construction of conformal mappings see also [[#References|[a1]]]. A good account of the theory of conformal mapping in the plane is given in the classics [[#References|[a2]]], [[#References|[a3]]], [[#References|[a7]]], in which also a number of special mappings are given.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Gaier,   "Konstruktive Methoden der konformen Abbildung" , Springer  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Z. Nehari,   "Conformal mapping" , Dover, reprint  (1952)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Tsuji,   "Potential theory in modern function theory" , Chelsea, reprint  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Cohn,   "Conformal mapping on Riemann surfaces" , Dover, reprint  (1980)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Kober,   "Dictionary of conformal representation" , Dover, reprint  (1952)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Courant,   "Dirichlet's principle, conformal mapping and minimal surfaces" , Wiley (Interscience)  (1950)  (With appendix by M. Schiffer: Some recent developments in the theory of conformal mapping)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Bieberbach,   "Conformal mapping" , Chelsea, reprint  (1964)  (Translated from German)</TD></TR></table>