Complex analysis

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Indirectly conformal mapping

An indirectly conformal mapping, sometimes called an anticonformal mapping, is a mapping that reverses all angles, whereas an isogonal mapping can reverse some angles and preserve others.For example, if is a conformal map, then is an indirectly conformal map, and is an isogonal mapping.


The symbol used by engineers and some physicists to denote i, the imaginary number . is probably preferred over because the symbol (or ) is commonly used to denote current.

Real axis

The real axis is the line in the complex plane corresponding to zero imaginary part, . Every real number corresponds to a unique point on the real axis.

Residue theorem

An analytic function whose Laurent series is given by(1)can be integrated term by term using a closed contour encircling ,(2)(3)The Cauchy integral theorem requires thatthe first and last terms vanish, so we have(4)where is the complex residue. Using the contour gives(5)so we have(6)If the contour encloses multiple poles, then the theorem gives the general result(7)where is the set of poles contained inside the contour. This amazing theorem therefore says that the value of a contour integral for any contour in the complex plane depends only on the properties of a few very special points inside the contour.The diagram above shows an example of the residue theorem applied to the illustrated contour and the function(8)Only the poles at 1 and are contained in the contour, which have residues of 0 and 2, respectively. The values of the contour integral is therefore given by(9)..

Principal part

If a function has a pole at , then the negative power part(1)of the Laurent series of about (2)is called the principal part of at . For example, the principal part of(3)is (Krantz 1999, pp. 46-47).

Complex residue

The constant in the Laurent series(1)of about a point is called the residue of . If is analytic at , its residue is zero, but the converse is not always true (for example, has residue of 0 at but is not analytic at ). The residue of a function at a point may be denoted . The residue is implemented in the Wolfram Language as Residue[f, z, z0].Two basic examples of residues are given by and for .The residue of a function around a point is also defined by(2)where is counterclockwise simple closed contour, small enough to avoid any other poles of . In fact, any counterclockwise path with contour winding number 1 which does not contain any other poles gives the same result by the Cauchy integral formula. The above diagram shows a suitable contour for which to define the residue of function, where the poles are indicated as black dots.It is more natural to consider the residue of a meromorphic one-form because it is independent of the choice of coordinate. On a Riemann..

Del bar operator

The operator is defined on a complex manifold, and is called the 'del bar operator.' The exterior derivative takes a function and yields a one-form. It decomposes as(1)as complex one-forms decompose into complexform of type(2)where denotes the direct sum. More concretely, in coordinates ,(3)and(4)These operators extend naturally to forms of higher degree. In general, if is a -complex form, then is a -form and is a -form. The equation expresses the condition of being a holomorphic function. More generally, a -complex form is called holomorphic if , in which case its coefficients, as written in a coordinate chart, are holomorphic functions.The del bar operator is also well-defined on bundle sections of a holomorphic vector bundle. The reason is because a change in coordinates or trivializations is holomorphic...

Zalcman's lemma

Let be a family of meromorphic functions on the unit disk which are not normal at 0. Then there exist sequences in , , , and a nonconstant function meromorphic in the plane such thatlocally and uniformly (in the spherical sense) in the complex plane (Schwick 2000), where and .

Montel's theorem

Let be an analytic function of , regular in the half-strip defined by and . If is bounded in and tends to a limit as for a certain fixed value of between and , then tends to this limit on every line in , and uniformly for .

Complex rotation

A complex rotation is a map of the form , where is a real number, which corresponds to counterclockwise rotation by radians about the origin of points the complex plane.

Lusin area integral

If is a domain and is a one-to-one analytic function, then is a domain, and(Krantz 1999, p. 150).

Logarithmic branch point

A branch point whose neighborhood of values wrap around an infinite number of times as their complex arguments are varied. The point under the function is therefore a logarithmic branch point. Logarithmic branch points are equivalent to logarithmic singularities.

Complex magnification

A complex magnification is a map of the form , where is a positive real number, which corresponds to magnification about the origin of points in the complex plane by the factor if is greater than 1, or shrinking by a factor if is less than 1.

Complex form

The differential forms on decompose into forms of type , sometimes called -forms. For example, on , the exterior algebra decomposes into four types:(1)(2)where , , and denotes the direct sum. In general, a -form is the sum of terms with s and s. A -form decomposes into a sum of -forms, where .For example, the 2-forms on decompose as(3)(4)The decomposition into forms of type is preserved by holomorphic functions. More precisely, when is holomorphic and is a -form on , then the pullback is a -form on .Recall that the exterior algebra is generated by the one-forms, by wedge product and addition. Then the forms of type are generated by(5)The subspace of the complex one-forms can be identified as the -eigenspace of the almost complex structure , which satisfies . Similarly, the -eigenspace is the subspace . In fact, the decomposition of determines the almost complex structure on .More abstractly, the forms into type are a group representation of , where..

Complex conjugate

The complex conjugate of a complex number is defined to be(1)The conjugate matrix of a matrix is the matrix obtained by replacing each element with its complex conjugate, (Arfken 1985, p. 210).The complex conjugate is implemented in the WolframLanguage as Conjugate[z].Note that there are several notations in common use for the complex conjugate. Applied physics and engineering texts tend to prefer , while most modern math and theoretical physics texts favor . Unfortunately, the notation is also commonly used to denote adjoint operators matrices. Because of these mutually contradictory conventions, care is needed when consulting the literature. In this work, is used to denote the complex conjugate.Common notational conventions for complex conjugate are summarized in the table below.notationreferencesThis work; Abramowitz and Stegun (1972, p. 16), Anton (2000, p. 528), Harris and Stocker (1998, p. 21),..

Complex analysis

Complex analysis is the study of complex numbers together with their derivatives, manipulation, and other properties. Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems. Contour integration, for example, provides a method of computing difficult integrals by investigating the singularities of the function in regions of the complex plane near and between the limits of integration.The key result in complex analysis is the Cauchy integral theorem, which is the reason that single-variable complex analysis has so many nice results. A single example of the unexpected power of complex analysis is Picard's great theorem, which states that an analytic function assumes every complex number, with possibly one exception, infinitely often in any neighborhood of an essential singularity!A fundamental result of complex analysis is the Cauchy-Riemann..

Symmetric points

Two points and are symmetric with respect to a circle or straight line if all circles and straight lines passing through and are orthogonal to . Möbius transformations preserve symmetry. Let a straight line be given by a point and a unit vector , thenwhere is the complex conjugate. Let a circle be given by center and radius , then

Köbe function

Min Max Re Im The function(1)defined on the unit disk . For , the Köbe function is a schlicht function(2)with for all (Krantz 1999, p. 149). For ,(3)illustrated above.

Carlson's theorem

If is regular and of the form where , for , and if for , 1, ..., then is identically zero.

Branch point

A branch point of an analytic function is a point in the complex plane whose complex argument can be mapped from a single point in the domain to multiple points in the range. For example, consider the behavior of the point under the power function(1)for complex non-integer , i.e., with . Writing and taking from 0 to gives(2)(3)so the values of at and are different, despite the fact that they correspond to the same point in the domain.Branch points whose neighborhood of values wrap around the range a finite number of times as varies from 0 to correspond to the point under functions of the form and are called algebraic branch points of order . A branch point whose neighborhood of values wraps around an infinite number of times occurs at the point under the function and is called a logarithmic branch point. Logarithmic branch points are equivalent to logarithmic singularities.Pinch points are also called branch points.It should be noted that the endpoints..

Schwarz's lemma

Let be analytic on the unit disk, and assume that 1. for all and 2. . Then and .If either for some or if , then is a rotation, i.e., for some complex constant with .

Branch cut

A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analytic multivalued function is discontinuous. For convenience, branch cuts are often taken as lines or line segments. Branch cuts (even those consisting of curves) are also known as cut lines (Arfken 1985, p. 397), slits (Kahan 1987), or branch lines.For example, consider the function which maps each complex number to a well-defined number . Its inverse function , on the other hand, maps, for example, the value to . While a unique principal value can be chosen for such functions (in this case, the principal square root is the positive one), the choices cannot be made continuous over the whole complex plane. Instead, lines of discontinuity must occur. The most common approach for dealing with these discontinuities is the adoption of so-called branch cuts. In general, branch cuts are not unique, but are instead chosen by convention..

Schlicht function

An analytic function on the unit disk is called schlicht if 1. is one-to-one, 2. , and 3. , in which case it is written . Schlicht functions have power series of the form


In complex analysis, a branch (also called a sheet) is a portion of the range of a multivalued function over which the function is single-valued. Combining all the sheets gives the full structure of the function. It is often convenient to choose a particular branch of a function to work with, and this choice is often designated the "principal branch" (or "principal sheet").In graph theory, a branch at a point in a tree is a maximal subtree containing as an endpoint (Harary 1994, p. 35).


A hyperfunction, discovered by Mikio Sato in 1958, is defined as a pair of holomorphic functions which are separated by a boundary . If is taken to be a segment on the real-line, then f is defined on the open region below the boundary and is defined on the open region above the boundary. A hyperfunction defined on gamma is the "jump" across the boundary from to .This pair forms an equivalence class of pairs of holomorphic functions , where is a holomorphic function defined on the open region , comprised of both and .Hyperfunctions can be shown to satisfy(1)(2)There is no general product between hyperfunctions, but the product of a hyperfunction by a holomorphic function can be expressed as(3)A standard holomorphic function can also be defined as a hyperfunction,(4)The Heaviside step function and the delta function can be defined as the hyperfunctions(5)(6)..

Blaschke product

A Blaschke product is an expression of the formwhere is a nonnegative integer and is the complex conjugate.

Riemann removable singularity theorem

Let be analytic and bounded on a punctured open disk , then exists, and the function defined by (1)is analytic.

Holomorphic function

A synonym for analytic function, regular function, differentiable function, complex differentiable function, and holomorphic map (Krantz 1999, p. 16). The word derives from the Greek (holos), meaning "whole," and (morphe), meaning "form" or "appearance."Many mathematicians prefer the term "holomorphic function" (or "holomorphic map") to "analytic function" (Krantz 1999, p. 16), while "analytic" appears to be in widespread use among physicists, engineers, and in some older texts (Morse and Feshbach 1953, pp. 356-374; Knopp 1996, pp. 83-111; Whittaker and Watson 1990, p. 83).

Blaschke factorization

Let be a bounded analytic function on vanishing to order at 0 and let be its other zeros, listed with multiplicities. Thenwhere is a bounded analytic function on , is zerofree, is the complex conjugate, and

Blaschke factor

If is a point in the open unit disk, then the Blaschke factor is defined bywhere is the complex conjugate of . Blaschke factors allow the manipulation of the zeros of a holomorphic function analogously to factors of for complex polynomials (Krantz 1999, p. 117).

Regular point

If is analytic on a domain , then a point on the boundary is called regular if extends to be an analytic function on an open set containing and also the point (Krantz 1999, p. 119).

Blaschke condition

If (with possible repetitions) satisfieswhere is the unit open disk, and no , then there is a bounded analytic function on which has zero set consisting precisely of the s, counted according to their multiplicities. More specifically, the infinite productwhere is a Blaschke factor and is the complex conjugate, converges uniformly on compact subsets of to a bounded analytic function .

Bergman kernel

A Bergman kernel is a function of a complex variable with the "reproducing kernel" property defined for any domain in which there exist nonzero analytic functions of class with respect to the Lebesgue measure .

Principle of permanence

In its simplest form, the principle of permanence states that, given any analytic function defined on an open (and connected) set of the complex numbers , and a convergent sequence which along with its limit belongs to , such that for all , then is uniformly zero on .This is easily proved by showing that the Taylor series of about must have all its coefficients equal to 0.The principle of permanence has wide-ranging consequences. For example, if and are analytic functions defined on , then any functional equation of the formthat is true for all in a closed subset of having a limit point in (e.g., a nonempty open subset of ) must be true for all in .

Automorphic function

An automorphic function of a complex variable is one which is analytic (except for poles) in a domain and which is invariant under a countably infinite group of linear fractional transformations (also known as Möbius transformations)Automorphic functions are generalizations of trigonometricfunctions and elliptic functions.

Principal branch

The principal branch of an analytic multivalued function, also called a principal sheet, is a single-valued "slice" (i.e., branch) of the function chosen that is for convenience in referring to a specific canonical value (a so-called principal value) of the function for each complex .For example, the principal branch of the natural logarithm, sometimes denoted , is the one for which , and hence is equal to the value for all (Knopp 1996, p. 111). The value of a function on its principal branch is known as its principal value. All values of then consist ofwith , ..., with the principal branch corresponding to . Since has only a single branch point, all branches can be plotted to give the entire Riemann surface.

Analytic function

A complex function is said to be analytic on a region if it is complex differentiable at every point in . The terms holomorphic function, differentiable function, and complex differentiable function are sometimes used interchangeably with "analytic function" (Krantz 1999, p. 16). Many mathematicians prefer the term "holomorphic function" (or "holomorphic map") to "analytic function" (Krantz 1999, p. 16), while "analytic" appears to be in widespread use among physicists, engineers, and in some older texts (e.g., Morse and Feshbach 1953, pp. 356-374; Knopp 1996, pp. 83-111; Whittaker and Watson 1990, p. 83).If a complex function is analytic on a region , it is infinitely differentiable in . A complex function may fail to be analytic at one or more points through the presence of singularities, or along lines or line segments through the presence..

Dispersion relation

Any pair of equations giving the real part of a function as an integral of its imaginary part and the imaginary part as an integral of its real part. Dispersion relationships imply causality in physics. Let(1)then(2)(3)where denotes the Cauchy principal value and and are Hilbert transforms of each other. If the complex function is symmetric such that , then(4)(5)

Algebraic branch point

A branch point whose neighborhood of values wrap around the range a finite number of times as their complex arguments varies from 0 to a multiple of is called an algebraic branch point of order . Such points correspond to the point under functions of the form .Formally, an algebraic branch point is a singular boundary point of one sheet of a multivalued function about which a finite number of distinct sheets hang together like the surface for at the origin and for which the domain of values affixed to these sheets in a neighborhood of , which can be developed in a series of the formis such that only a finite number (or zero) negative power of appear in the expansion (Knopp 1996, Part II, p. 143).

Wright function

The entire functionwhere and , named after the British mathematician E. M. Wright.

Entire function

If a complex function is analytic at all finite points of the complex plane , then it is said to be entire, sometimes also called "integral" (Knopp 1996, p. 112).Any polynomial is entire.Examples of specific entire functions are given in the following table.functionsymbolAiry functions, Airy function derivatives, Anger functionBarnes G-functionbeiberBessel function of the first kindBessel function of the second kindBeurling's functioncosinecoversineDawson's integralerferfcerfiexponential functionFresnel integrals, gamma function reciprocalgeneralized hypergeometric functionhaversinehyperbolic cosinehyperbolic sineJacobi elliptic functions, , , , , , , , , , , Jacobi theta functionsJacobi theta function derivativesMittag-Leffler functionmodified Struve functionNeville theta functions, , , Shisinesine integralspherical Bessel function of the first kindStruve functionversineWeber..

Dawson's integral

Dawson's integral (Abramowitz and Stegun 1972, pp. 295 and 319), also sometimes called Dawson's function, is the entire function given by the integral(1)(2)where is erfi, that arises in computation of the Voigt lineshape (Harris 1948, Hummer 1963, Sajo 1993, Lether 1997), in heat conduction, and in the theory of electrical oscillations in certain special vacuum tubes (McCabe 1974). It is commonly denoted (McCabe 1974; Coleman 1987; Milone and Milone 1988; Sajo 1993; Lether 1997; Press et al. 2007, p. 302), although Spanier and Oldham (1987) denote it by .Dawson's integral is implemented in the WolframLanguage as DawsonF[z].It is an odd function, so(3)Its derivative is(4)and its indefinite integral is(5)where is a generalized hypergeometric function.It is the particular solution to the differential equation(6)(McCabe 1974).Its Maclaurin series is given by(7)(8)(OEIS A122803 and A001147).If has the asymptotic series(9)It..

Sine integral

Min Max Min Max Re Im The most common "sine integral" is defined as(1) is the function implemented in the Wolfram Language as the function SinIntegral[z]. is an entire function.A closed related function is defined by(2)(3)(4)(5)where is the exponential integral, (3) holds for , and(6)The derivative of is(7)where is the sinc function and the integral is(8)A series for is given by(9)(Havil 2003, p. 106).It has an expansion in terms of sphericalBessel functions of the first kind as(10)(Harris 2000).The half-infinite integral of the sinc functionis given by(11)To compute the integral of a sine function times a power(12)use integration by parts. Let(13)(14)so(15)Using integration by parts again,(16)(17)(18)Letting , so(19)General integrals of the form(20)are related to the sinc function and can be computedanalytically...

Picard's little theorem

Any entire analytic function whose range omits two points must be a constant function.Of course, an entire function that omits a single point from its range need not be a constant, as illustrated by the function , which is entire but omits the point from its range.

Function order

The infimum of all number for whichholds for all and an entire function, is called the order of , denoted (Krantz 1999, p. 121).

Fresnel integrals

There are a number of slightly different definitions of the Fresnel integrals. In physics, the Fresnel integrals denoted and are most often defined by(1)(2)so(3)(4)These Fresnel integrals are implemented in the Wolfram Language as FresnelC[z] and FresnelS[z]. and are entire functions. Min Max Re Im Min Max Re Im The and integrals are illustrated above in the complex plane.They have the special values(5)(6)(7)and(8)(9)(10)An asymptotic expansion for gives(11)(12)Therefore, as , and . The Fresnel integrals are sometimes alternatively defined as(13)(14)Letting so , and (15)(16)In this form, they have a particularly simple expansion in terms of sphericalBessel functions of the first kind. Using(17)(18)(19)where is a spherical Bessel function of the second kind(20)(21)(22)(23)(24)Related functions , , , and are defined by(25)(26)(27)(28)..

Finite order

An entire function is said to be of finite order if there exist numbers such thatfor all . The infimum of all numbers for which this inequality holds is called the function order of , denoted .

Part metric

A metric defined bywhere denotes the positive harmonic functions on a domain. The part metric is invariant under conformal maps for any domain.

Punctured plane

The complex plane with the origin removed, i.e., . The punctured plane is sometimes denoted (although this notation conflicts with that for the Riemann sphere C-*, or extended complex plane). It is both a Lie group and an Abelian variety.

Variation of argument

Let denote the change in the complex argument of a function around a contour . Also let denote the number of roots of in and denote the sum of the orders of all poles of lying inside . Then(1)For example, the plots above shows the argument for a small circular contour centered around for a function of the form (which has a single pole of order and no roots in ) for , 2, and 3.Note that the complex argument must change continuously, so any "jumps" that occur as the contour crosses branch cuts must be taken into account.To find in a given region , break into paths and find for each path. On a circular arc(2)let be a polynomial of degree . Then(3)(4)Plugging in gives(5)So as ,(6)(7)and(8)For a real segment ,(9)For an imaginary segment ,(10)

Morera's theorem

If is continuous in a region and satisfiesfor all closed contours in , then is analytic in .Morera's theorem does not require simple connectedness, which can be seen from the following proof. Let be a region, with continuous on , and let its integrals around closed loops be zero. Pick any point , and pick a neighborhood of . Construct an integral of ,Then one can show that , and hence is analytic and has derivatives of all orders, as does , so is analytic at . This is true for arbitrary , so is analytic in .It is, in fact, sufficient to require that the integrals of around triangles be zero, but this is a technical point. In this case, the proof is identical except must be constructed by integrating along the line segment instead of along an arbitrary path.

Jordan's lemma

Jordan's lemma shows the value of the integral(1)along the infinite upper semicircle and with is 0 for "nice" functions which satisfy . Thus, the integral along the real axis is just the sum of complex residues in the contour.The lemma can be established using a contour integral that satisfies(2)To derive the lemma, write(3)(4)(5)and define the contour integral(6)Then(7)(8)(9)Now, if , choose an such that , so(10)But, for ,(11)so(12)(13)(14)As long as , Jordan's lemma(15)then follows.

Contour winding number

The winding number of a contour about a point , denoted , is defined byand gives the number of times curve passes (counterclockwise) around a point. Counterclockwise winding is assigned a positive winding number, while clockwise winding is assigned a negative winding number. The winding number is also called the index, and denoted .The contour winding number was part of the inspiration for the idea of the Brouwer degree between two compact, oriented manifolds of the same dimension. In the language of the degree of a map, if is a closed curve (i.e., ), then it can be considered as a function from to . In that context, the winding number of around a point in is given by the degree of the mapfrom the circle to the circle.

Contour integration

Contour integration is the process of calculating the values of a contour integral around a given contour in the complex plane. As a result of a truly amazing property of holomorphic functions, such integrals can be computed easily simply by summing the values of the complex residues inside the contour.Let and be polynomials of polynomial degree and with coefficients , ..., and , ..., . Take the contour in the upper half-plane, replace by , and write . Then(1)Define a path which is straight along the real axis from to and make a circular half-arc to connect the two ends in the upper half of the complex plane. The residue theorem then gives(2)(3)(4)where denotes the complex residues. Solving,(5)Define(6)(7)(8)(9)and set(10)then equation (9) becomes(11)Now,(12)for . That means that for , or , , so(13)for . Apply Jordan's lemma with . We must have(14)so we require .Then(15)for and . Since this must hold separately for real and imaginary parts, this..


A path in the complex plane over which contour integration is performed to compute a contour integral. When choosing a contour to evaluate an integral on the real line, a contour is generally chosen based on the range of integration and the position of poles in the complex plane. For example, for an integral from to along the real axis, the contour at left could be chosen if the function had no poles on the real line, and the middle contour could be chosen if it had a pole at the origin. To perform an integral over the positive real axis from 0 to for a function with a pole at 0, the contour at right could be chosen.

Cauchy integral theorem

If is analytic in some simply connected region , then(1)for any closed contour completely contained in . Writing as(2)and as(3)then gives(4)(5)From Green's theorem,(6)(7)so (◇) becomes(8)But the Cauchy-Riemann equations requirethat(9)(10)so(11)Q.E.D.For a multiply connected region,(12)

Cauchy integral formula

Cauchy's integral formula states that(1)where the integral is a contour integral along the contour enclosing the point .It can be derived by considering the contour integral(2)defining a path as an infinitesimal counterclockwise circle around the point , and defining the path as an arbitrary loop with a cut line (on which the forward and reverse contributions cancel each other out) so as to go around . The total path is then(3)so(4)From the Cauchy integral theorem, the contour integral along any path not enclosing a pole is 0. Therefore, the first term in the above equation is 0 since does not enclose the pole, and we are left with(5)Now, let , so . Then(6)(7)But we are free to allow the radius to shrink to 0, so(8)(9)(10)(11)giving (1).If multiple loops are made around the point , then equation (11) becomes(12)where is the contour winding number.A similar formula holds for the derivatives of ,(13)(14)(15)(16)(17)Iterating again,(18)Continuing..

Argument principle

If is meromorphic in a region enclosed by a contour , let be the number of complex roots of in , and be the number of poles in , with each zero and pole counted as many times as its multiplicity and order, respectively. ThenDefining and gives

Conformal radius

Let be a simply connected compact set in the complex plane. By the Riemann mapping theorem, there is a unique analytic function(1)for that maps the exterior of the unit disk conformally onto the exterior of and takes to . The number is called the conformal radius of and is called the conformal center of .The function carries interesting information about the set . For instance, is equal to the logarithmic capacity of and(2)where the equality holds iff is a segment of length . The Green's function associated to Laplace's equation for the exterior of with respect to is given by(3)for .

Transfinite diameter

Letbe an analytic function, regular and univalent for , that maps conformally onto the region preserving the point at infinity and its direction. Then the function is uniquely determined and is called the transfinite diameter, sometimes also known as Robin's constant or the capacity of .

Nonconformal map

Let be a path in , , and and be the tangents to the curves and at and . If there is an such that(1)(2)for all (or, equivalently, if has a zero of order ), then(3)(4)so the complex argument is(5)As , and ,(6)

Conformal mapping

A conformal mapping, also called a conformal map, conformal transformation, angle-preserving transformation, or biholomorphic map, is a transformation that preserves local angles. An analytic function is conformal at any point where it has a nonzero derivative. Conversely, any conformal mapping of a complex variable which has continuous partial derivatives is analytic. Conformal mapping is extremely important in complex analysis, as well as in many areas of physics and engineering.A mapping that preserves the magnitude of angles, but not their orientation is called an isogonal mapping (Churchill and Brown 1990, p. 241).Several conformal transformations of regular grids are illustrated in the first figure above. In the second figure above, contours of constant are shown together with their corresponding contours after the transformation. Moon and Spencer (1988) and Krantz (1999, pp. 183-194) give tables of conformal..

Logarithmic capacity

The logarithmic capacity of a compact set in the complex plane is given by(1)where(2)and runs over each probability measure on . The quantity is called the Robin's constant of and the set is said to be polar if or equivalently, .The logarithmic capacity coincides with the transfinite diameter of ,(3)If is simply connected, the logarithmic capacity of is equal to the conformal radius of . Tables of logarithmic capacities have been calculated (e.g., Rumely 1989).

Cayley transform

The linear fractional transformationthat maps the upper half-plane conformally onto the unit disk .

Linear fractional transformation

A transformation of the form(1)where , , , and(2)is a conformal mapping called a linear fractional transformation. The transformation can be extended to the entire extended complex plane by defining(3)(4)(Apostol 1997, p. 26). The linear fractional transformation is linear in both and , and analytic everywhere except for a simple pole at .Kleinian groups are the most general case of discrete groups of linear fractional transformations in the complex plane .Every linear fractional transformation except has one or two fixed points. The linear fractional transformation sends circles and lines to circles or lines. Linear fractional transformations preserve symmetry. The cross ratio is invariant under a linear fractional transformation. A linear fractional transformation is a composition of translations, rotations, magnifications, and inversions.To determine a particular linear fractional transformation, specify the..

Carathéodory's theorem

If and are bounded domains, , are Jordan curves, and is a conformal mapping, then (respectively, ) extends one-to-one and continuously to (respectively, ).

Isogonal mapping

An isogonal mapping is a transformation that preserves the magnitudes of local angles, but not their orientation. A few examples are illustrated above.A conformal mapping is an isogonal mapping that also preserves the orientations of local angles. If is a conformal mapping, then is isogonal but not conformal. This is due to the fact that complex conjugation is not an analytic function.

Riemann mapping theorem

Let be a point in a simply connected region , where is the complex plane. Then there is a unique analytic function mapping one-to-one onto the disk such that and . The corollary guarantees that any two simply connected regions except (the Euclidean plane) can be mapped conformally onto each other.

Ahlfors five island theorem

Let be a transcendental meromorphic function, and let , , ..., be five simply connected domains in with disjoint closures (Ahlfors 1932). Then there exists and, for any , a simply connected domain such that is a conformal mapping of onto . If has only finitely many poles, then "five" may be replaced by "three" (Ahlfors 1933).

Complex number paradox

An improper use of the symbol for the imaginary unit leads to the apparent proof of a false statement.(1)(2)(3)(4)(5)The reason for the fallacy is that is not an ordinary (real) square root, hence the rule for computing the quotient of radicals does not apply to it.

Complex number

The complex numbers are the field of numbers of the form , where and are real numbers and i is the imaginary unit equal to the square root of , . When a single letter is used to denote a complex number, it is sometimes called an "affix." In component notation, can be written . The field of complex numbers includes the field of real numbers as a subfield.The set of complex numbers is implemented in the Wolfram Language as Complexes. A number can then be tested to see if it is complex using the command Element[x, Complexes], and expressions that are complex numbers have the Head of Complex.Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept. For example, John Wallis wrote, "These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative..

Complex multiplication

Two complex numbers and are multiplied as follows:(1)(2)(3)In component form,(4)(Krantz 1999, p. 1). The special case of a complex number multiplied by a scalar is then given by(5)Surprisingly, complex multiplication can be carried out using only three real multiplications, , , and as(6)(7)Complex multiplication has a special meaning for ellipticcurves.

Imaginary unit

The imaginary number , i.e., the square root of . The imaginary unit is denoted and commonly referred to as "i." Although there are two possible square roots of any number, the square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point and can then be distinguished. Since either choice is possible, there is no ambiguity in defining as "the" square root of . In the Wolfram Language, the imaginary unit is implemented as I.

Complex exponentiation

A complex number may be taken to the power of another complex number. In particular, complex exponentiation satisfies(1)where is the complex argument. Written explicitly in terms of real and imaginary parts,(2)An explicit example of complex exponentiation is given by(3)A complex number taken to a complex number can be real. In fact, the famous example(4)shows that the power of the purely imaginary to itself is real. Min Max Re Im In fact, there is a family of values such that is real, as can be seen by writing(5)This will be real when , i.e., for(6)for an integer. For positive , this gives roots or(7)where is the Lambert W-function. For , this simplifies to(8)For , 2, ..., these give the numeric values 1, 2.92606 (OEIS A088928), 4.30453, 5.51798, 6.63865, 7.6969, ......

Real part

The real part of a complex number is the real number not multiplying i, so . In terms of itself,where is the complex conjugate of . The real part is implemented in the Wolfram Language as Re[z].A nonzero complex number with zero real part is called an imaginary number or sometimes, for emphasis, a purely imaginary number.

Complex division

The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator, for example, with and , is given by(1)(2)(3)(4)(5)where denotes the complex conjugate. In component notation with ,(6)

Imaginary part

The imaginary part of a complex number is the real number multiplying i, so . In terms of itself,where is the complex conjugate of . The imaginary part is implemented in the Wolfram Language as Im[z].

Complex argument

Min Max Re Im A complex number may be represented as(1)where is a positive real number called the complex modulus of , and (sometimes also denoted ) is a real number called the argument. The argument is sometimes also known as the phase or, more rarely and more confusingly, the amplitude (Derbyshire 2004, pp. 180-181 and 376).The complex argument of a number is implemented in the Wolfram Language as Arg[z].The complex argument can be computed as(2)Here, , sometimes also denoted , corresponds to the counterclockwise angle from the positive real axis, i.e., the value of such that and . The special kind of inverse tangent used here takes into account the quadrant in which lies and is returned by the FORTRAN command ATAN2(y, x) and the Wolfram Language function ArcTan[x, y], and is often (including by the Wolfram Language function Arg) restricted to the range . In the degenerate case when ,(3)Special values of the complex argument include(4)(5)(6)(7)(8)From..

Purely imaginary number

A complex number is said to be purely imaginary if it has no real part, i.e., . The term is often used in preference to the simpler "imaginary" in situations where can in general assume complex values with nonzero real parts, but in a particular case of interest, the real part is identically zero.

Imaginary number

Although Descartes originally used the term "imaginary number" to refer to what is today known as a complex number, in standard usage today, "imaginary number" means a complex number that has zero real part (i.e., such that ). For clarity, such numbers are perhaps best referred to as purely imaginary numbers.A (purely) imaginary number can be written as a real number multiplied by the "imaginary unit" i (equal to the square root ), i.e., in the form .In the novel The Da Vinci Code, the character Robert Langdon jokes that character Sophie Neveu "believes in the imaginary number because it helps her break code" (Brown 2003, p. 351). In Isaac Asimov's short story "The Imaginary" (1942), eccentric psychologist Tan Porus explains the behavior of a mysterious species of squid by using imaginary numbers in the equations which describe its psychology. The anthology Imaginary Numbers:..

Complex addition

Two complex numbers and are added together componentwise,In component form,(Krantz 1999, p. 1).


The representation, beloved of engineers and physicists, of a complexnumber in terms of a complex exponential(1)where i (called j by engineers) is the imaginary number and the complex modulus and complex argument (also called phase) are(2)(3)Here, (sometimes also denoted ) is called the complex argument or the phase. It corresponds to the counterclockwise angle from the positive real axis, i.e., the value of such that and . The special kind of inverse tangent used here takes into account the quadrant in which lies and is returned by the FORTRAN command ATAN2(Y,X) and the Wolfram Language function ArcTan[x, y], and is often restricted to the range . In the degenerate case when ,(4)It is trivially true that(5)Now consider a scalar function . Then(6)(7)(8)(9)where is the complex conjugate. Look at the time averages of each term,(10)(11)(12)(13)(14)(15)(16)(17)(18)Therefore,(19)Consider now two scalar functions(20)(21)Then(22)(23)(24)(25)(26)(27)In..

Gaussian integer

A Gaussian integer is a complex number where and are integers. The Gaussian integers are members of the imaginary quadratic field and form a ring often denoted , or sometimes (Hardy and Wright 1979, p. 179). The sum, difference, and product of two Gaussian integers are Gaussian integers, but only if there is an such that(1)(Shanks 1993).Gaussian integers can be uniquely factored in terms of other Gaussian integers (known as Gaussian primes) up to powers of and rearrangements.The units of are and .One definition of the norm of a Gaussian integer is its complexmodulus(2)Another common definition (e.g., Herstein 1975; Hardy and Wright 1979, p. 182; Artin 1991; Dummit and Foote 2004) defines the norm of a Gaussian integer to be(3)the square of the above quantity. (Note that the Gaussian integers form a Euclidean ring, which is what makes them particularly of interest, only under the latter definition.) Because of the two possible definitions,..

Extended complex plane

The extended complex plane is the name gives to the complex plane with a point at infinity attached: , where denotes complex infinity. It is also called the Riemann sphere and is various denoted or .

Argand diagram

An Argand diagram is a plot of complex numbersas pointsin the complex plane using the x-axis as the real axis and y-axis as the imaginary axis. In the plot above, the dashed circle represents the complex modulus of and the angle represents its complex argument.While Argand (1806) is generally credited with the discovery, the Argand diagram (also known as the Argand plane) was actually described by C. Wessel prior to Argand. Historically, the geometric representation of a complex number as a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, this visualization helped "imaginary" and "complex" numbers become accepted in mainstream mathematics as a natural extension to negative numbers along the real line...

Gaussian prime

Gaussian primes are Gaussian integers satisfying one of the following properties. 1. If both and are nonzero then, is a Gaussian prime iff is an ordinary prime. 2. If , then is a Gaussian prime iff is an ordinary prime and . 3. If , then is a Gaussian prime iff is an ordinary prime and . The above plot of the complex plane shows the Gaussianprimes as filled squares.The primes which are also Gaussian primes are 3, 7, 11, 19, 23, 31, 43, ... (OEIS A002145). The Gaussian primes with are given by , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 3, , , , , , , , , .The numbers of Gaussian primes with complex modulus (where the definition has been used) for , 1, ... are 0, 100, 4928, 313752, ... (OEIS A091134).The cover of Bressoud and Wagon (2000) shows an illustration of the distributionof Gaussian primes in the complex plane.As of 2009, the largest known Gaussian prime, found in Sep. 2006, is , whose real and imaginary parts both have decimal digits and whose squared..

Contour integral

An integral obtained by contour integration. The particular path in the complex plane used to compute the integral is called a contour.As a result of a truly amazing property of holomorphic functions, a closed contour integral can be computed simply by summing the values of the complex residues inside the contour.Watson (1966 p. 20) uses the notation to denote the contour integral of with contour encircling the point once in a counterclockwise direction.Renteln and Dundes (2005) give the following (bad) mathematical joke about contour integrals:Q: What's the value of a contour integral around Western Europe? A: Zero, because all the Poles are in Eastern Europe.

Kms condition

The Kubo-Martin-Schwinger (KMS) condition is a kind of boundary-value condition which naturally emerges in quantum statistical mechanics and related areas.Given a quantum system with finite dimensional Hilbert space , define the function as(1)where is the imaginary unit and where is the Hamiltonian, i.e., the sum of the kinetic energies of all the particles in plus the potential energy of the particles associated with . Next, for any real number , define the thermal equilibrium as(2)where denotes the matrix trace. From and , one can define the so-called equilibrium correlation function where(3)whereby the KMS boundary condition says that(4)In particular, this identity relates to the state the values of the analytic function on the boundary of the strip(5)where here, denotes the imaginary part of and denotes the signum function applied to .In various literature, the KMS boundary condition is stated in sometimes-different contexts...

Jensen polynomial

Let be a real entire function of the form(1)where the s are positive and satisfy Turán's inequalities(2)for , 2, .... The Jensen polynomial associated with is then given by(3)where is a binomial coefficient.

Euler formula

The Euler formula, sometimes also called the Euler identity (e.g., Trott 2004, p. 174), states(1)where i is the imaginary unit. Note that Euler's polyhedral formula is sometimes also called the Euler formula, as is the Euler curvature formula. The equivalent expression(2)had previously been published by Cotes (1714).The special case of the formula with gives the beautiful identity(3)an equation connecting the fundamental numbers i, pi, e, 1, and 0 (zero), the fundamental operations , , and exponentiation, the most important relation , and nothing else. Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician (Derbyshire 2004, p. 202).The Euler formula can be demonstrated using a series expansion(4)(5)(6)It can also be demonstrated using a complex integral.Let(7)(8)(9)(10)(11)(12)so(13)(14)A mathematical joke asks, "How many..

Complex plane

The complex plane is the plane of complex numbers spanned by the vectors 1 and , where is the imaginary number. Every complex number corresponds to a unique point in the complex plane. The line in the plane with is the real line. The complex plane is sometimes called the Argand plane or Gauss plane, and a plot of complex numbers in the plane is sometimes called an Argand diagram.The complex plane together with the point at infinity is known as the Riemann sphere or extended complex plane and denoted or . However, the notation is also used to denote the punctured plane .

Absolute square

Min Max Min Max Re Im The absolute square of a complex number , also known as the squared norm, is defined as(1)where denotes the complex conjugate of and is the complex modulus.If the complex number is written , with and real, then the absolute square can be written(2)If is a real number, then (1) simplifies to(3)An absolute square can be computed in terms of and using the Wolfram Language command ComplexExpand[Abs[z]^2, TargetFunctions -> Conjugate].An important identity involving the absolute square is given by(4)(5)(6)If , then (6) becomes(7)(8)If , and , then(9)Finally,(10)(11)(12)

Monogenic function

Ifis the same for all paths in the complex plane, then is said to be monogenic at . Monogenic therefore essentially means having a single derivative at a point. Functions are either monogenic or have infinitely many derivatives (in which case they are called polygenic); intermediate cases are not possible.

Meromorphic function

A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities). A simpler definition states that a meromorphic function is a function of the formwhere and are entire functions with (Krantz 1999, p. 64).A meromorphic function therefore may only have finite-order, isolated poles and zeros and no essential singularities in its domain. A meromorphic function with an infinite number of poles is exemplified by on the punctured disk , where is the open unit disk.An equivalent definition of a meromorphic function is a complex analytic mapto the Riemann sphere.The word derives from the Greek (meros), meaning "part," and (morphe), meaning "form" or "appearance."..

Complex differentiable

Let and on some region containing the point . If satisfies the Cauchy-Riemann equations and has continuous first partial derivatives in the neighborhood of , then exists and is given byand the function is said to be complex differentiable (or, equivalently, analyticor holomorphic).A function can be thought of as a map from the plane to the plane, . Then is complex differentiable iff its Jacobian is of the format every point. That is, its derivative is given by the multiplication of a complex number . For instance, the function , where is the complex conjugate, is not complex differentiable.

Schwarz reflection principle

Suppose that is an analytic function which is defined in the upper half-disk . Further suppose that extends to a continuous function on the real axis, and takes on real values on the real axis. Then can be extended to an analytic function on the whole disk by the formulaand the values for reflected across the real axis are the reflections of across the real axis. It is easy to check that the above function is complex differentiable in the interior of the lower half-disk. What is remarkable is that the resulting function must be analytic along the real axis as well, despite no assumptions of differentiability.This is called the Schwarz reflection principle, and is sometimes also known as Schwarz's symmetric principle (Needham 2000, p. 257). The diagram above shows the reflection principle applied to a function defined for the upper half-disk (left figure; red) and its image (right figure; red). The function is real on the real axis, so it is possible..

Monodromy theorem

If a complex function is analytic in a disk contained in a simply connected domain and can be analytically continued along every polygonal arc in , then can be analytically continued to a single-valued analytic function on all of !

Function element

A function element is an ordered pair where is a disk and is an analytic function defined on . If is an open set, then a function element in is a pair such that .

Direct analytic continuation

If and are functions elements, then is a direct analytic continuation of if and and are equal on .

Analytic continuation

Analytic continuation (sometimes called simply "continuation") provides a way of extending the domain over which a complex function is defined. The most common application is to a complex analytic function determined near a point by a power series(1)Such a power series expansion is in general valid only within its radius of convergence. However, under fortunate circumstances (that are very fortunately also rather common!), the function will have a power series expansion that is valid within a larger-than-expected radius of convergence, and this power series can be used to define the function outside its original domain of definition. This allows, for example, the natural extension of the definition trigonometric, exponential, logarithmic, power, and hyperbolic functions from the real line to the entire complex plane . Similarly, analytic continuation can be used to extend the values of an analytic function across a branch..

Three circles theorem

The three circles theorem, also called Hadamard's three circles theorem (Edwards 2001, p. 187), states that if is an analytic function in the annulus , , and , , and are the maxima of on the three circles corresponding to , , and , respectively, then(Derbyshire 2004, p. 376).The theorem was first published by Hadamard in 1896, although without proof (Bohr and Landau 1913; Edwards 2001, p. 187).


Conjugation is the process of taking a complex conjugate of a complex number, complex matrix, etc., or of performing a conjugation move on a knot.Conjugation also has a meaning in group theory. Let be a group and let . Then, defines a homomorphism given byThis is a homomorphism becauseThe operation on given by is called conjugation by .Conjugation is an important construction in group theory. Conjugation defines a group action of a group on itself and this often yields useful information about the group. For example, this technique is how the Sylow Theorems are proven. More importantly, a normal subgroup of a group is a subgroup which is invariant under conjugation by any element. Normal groups are extremely important because they are the kernels of homomorphisms and it is possible to take the quotient of a group and one of its normal subgroups...

Hadamard factorization theorem

Let be an entire function of finite order and the zeros of , listed with multiplicity, then the rank of is defined as the least positive integer such that(1)Then the canonical Weierstrass product is given by(2)and has degree . The genus of is then defined as , and the Hadamard factorization theory states that an entire function of finite order is also of finite genus , and(3)

Principal value

The principal value of an analytic multivalued function is the single value chosen by convention to be returned for a given argument. Complex multivalued functions have multiple branches in the complex plane, with those corresponding to the principal values known as the principal branch. For example, the principal branch of the natural logarithm, sometimes denoted , is the one for which , and hence is equal to the value for all (Knopp 1996, p. 111). All values of then consist ofwith , ..., with the principal branch corresponding to . Since has only a single branch point, all branches can be plotted to give the Riemann surface.The term "principal value" also occurs in the theory of integration (e.g., Vladimirov 1971, p. 75), where it means something completely different and is more properly known as the Cauchy principal value. The Cauchy principal value of an integral is implemented in the Wolfram Language using the command..

Simple pole

A simple pole of an analytic function is a pole of order one. That is, is an analytic function at the pole . Alternatively, its principal part is for some . It is called simple because a function with a pole of order at can be written as the product of functions with simple poles at .


The word "pole" is used prominently in a number of very different branches of mathematics. Perhaps the most important and widespread usage is to denote a singularity of a complex function. In inversive geometry, the inversion pole is related to inverse points with respect to an inversion circle. The term "pole" is also used to denote the degenerate points and in spherical coordinates, corresponding to the north pole and south pole respectively. "All-poles method" is an alternate term for the maximum entropy method used in deconvolution. In triangle geometry, an orthopole is the point of concurrence certain perpendiculars with respect to a triangle of a given line, and a Simson line pole is similarly defined based on the Simson line of a point with respect to a triangle. In projective geometry, the perspector is sometimes known as the perspective pole.In complex analysis, an analytic function is said to have..

Weierstrass product theorem

Let any finite or infinite set of points having no finite limit point be prescribed, and associate with each of its points a definite positive integer as its order. Then there exists an entire function which has zeros to the prescribed orders at precisely the prescribed points, and is otherwise different from zero. Moreover, this function can be represented as a product from which one can read off again the positions and orders of the zeros. Furthermore, if is one such function, thenis the most general function satisfying the conditions of the problem, where denotes an arbitrary entire function.This theorem is also sometimes simply known as Weierstrass's theorem. A spectacularexample is given by the Hadamard product.

Riemann surface

A Riemann surface is a surface-like configuration that covers the complex plane with several, and in general infinitely many, "sheets." These sheets can have very complicated structures and interconnections (Knopp 1996, pp. 98-99). Riemann surfaces are one way of representing multiple-valued functions; another is branch cuts. The above plot shows Riemann surfaces for solutions of the equationwith , 3, 4, and 5, where is the Lambert W-function (M. Trott).The Riemann surface of the function field is the set of nontrivial discrete valuations on . Here, the set corresponds to the ideals of the ring of integers of over . ( consists of the elements of that are roots of monic polynomials over .) Riemann surfaces provide a geometric visualization of functions elements and their analytic continuations.Schwarz proved at the end of nineteenth century that the automorphism group of a compact Riemann surface of genus is finite,..

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