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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an
irregular singularity In mathematics, in the theory of ordinary differential equations in the complex plane \Complex, the points of \Complex are classified into ''ordinary points'', at which the equation's coefficients are analytic functions, and ''singular points'', at ...
. The term ''
confluent In geography, a confluence (also: ''conflux'') occurs where two or more flowing bodies of water join to form a single channel. A confluence can occur in several configurations: at the point where a tributary joins a larger river (main stem); o ...
'' refers to the merging of singular points of families of differential equations; ''confluere'' is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions: * Kummer's (confluent hypergeometric) function , introduced by , is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated
Kummer's function In mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm. Both are named for Ernst Kummer Ernst Eduard K ...
bearing the same name. * Tricomi's (confluent hypergeometric) function introduced by , sometimes denoted by , is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind. *
Whittaker function In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced W ...
s (for
Edmund Taylor Whittaker Sir Edmund Taylor Whittaker (24 October 1873 – 24 March 1956) was a British mathematician, physicist, and historian of science. Whittaker was a leading mathematical scholar of the early 20th-century who contributed widely to applied mathema ...
) are solutions to Whittaker's equation. *
Coulomb wave function In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of conflu ...
s are solutions to the Coulomb wave equation. The Kummer functions, Whittaker functions, and Coulomb wave functions are essentially the same, and differ from each other only by elementary functions and change of variables.


Kummer's equation

Kummer's equation may be written as: :z\frac + (b-z)\frac - aw = 0, with a regular singular point at and an irregular singular point at . It has two (usually)
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
solutions and . Kummer's function of the first kind is a
generalized hypergeometric series In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...
introduced in , given by: :M(a,b,z)=\sum_^\infty \frac =_1F_1(a;b;z), where: : a^=1, : a^=a(a+1)(a+2)\cdots(a+n-1)\, , is the
rising factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...
. Another common notation for this solution is . Considered as a function of , , or with the other two held constant, this defines an
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
of or , except when As a function of it is analytic except for poles at the non-positive integers. Some values of and yield solutions that can be expressed in terms of other known functions. See #Special cases. When is a non-positive integer, then Kummer's function (if it is defined) is a generalized
Laguerre polynomial In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only ...
. Just as the confluent differential equation is a limit of the hypergeometric differential equation as the singular point at 1 is moved towards the singular point at ∞, the confluent hypergeometric function can be given as a limit of the hypergeometric function :M(a,c,z) = \lim__2F_1(a,b;c;z/b) and many of the properties of the confluent hypergeometric function are limiting cases of properties of the hypergeometric function. Since Kummer's equation is second order there must be another, independent, solution. The indicial equation of the method of Frobenius tells us that the lowest power of a power series solution to the Kummer equation is either 0 or . If we let be :w(z)=z^v(z) then the differential equation gives :z^\frac+2(1-b)z^\frac-b(1-b)z^v + (b-z)\left ^\frac+(1-b)z^v\right- az^v = 0 which, upon dividing out and simplifying, becomes :z\frac+(2-b-z)\frac - (a+1-b)v = 0. This means that is a solution so long as is not an integer greater than 1, just as is a solution so long as is not an integer less than 1. We can also use the Tricomi confluent hypergeometric function introduced by , and sometimes denoted by . It is a combination of the above two solutions, defined by :U(a,b,z)=\fracM(a,b,z)+\fracz^M(a+1-b,2-b,z). Although this expression is undefined for integer , it has the advantage that it can be extended to any integer by continuity. Unlike Kummer's function which is an
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
of , usually has a singularity at zero. For example, if and then is asymptotic to as goes to zero. But see #Special cases for some examples where it is an entire function (polynomial). Note that the solution to Kummer's equation is the same as the solution , see #Kummer's transformation. For most combinations of real or complex and , the functions and are independent, and if is a non-positive integer, so doesn't exist, then we may be able to use as a second solution. But if is a non-positive integer and is not a non-positive integer, then is a multiple of . In that case as well, can be used as a second solution if it exists and is different. But when is an integer greater than 1, this solution doesn't exist, and if then it exists but is a multiple of and of In those cases a second solution exists of the following form and is valid for any real or complex and any positive integer except when is a positive integer less than : :M(a,b,z)\ln z+z^\sum_^\infty C_kz^k When ''a'' = 0 we can alternatively use: :\int_^z(-u)^e^u\mathrmu. When this is the
exponential integral In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. Definitions For real non-zero values of&n ...
. A similar problem occurs when is a negative integer and is an integer less than 1. In this case doesn't exist, and is a multiple of A second solution is then of the form: :z^M(a+1-b,2-b,z)\ln z+\sum_^\infty C_kz^k


Other equations

Confluent Hypergeometric Functions can be used to solve the Extended Confluent Hypergeometric Equation whose general form is given as: :z\frac +(b-z)\frac -\left(\sum_^M a_m z^m\right)w = 0 Note that for or when the summation involves just one term, it reduces to the conventional Confluent Hypergeometric Equation. Thus Confluent Hypergeometric Functions can be used to solve "most" second-order ordinary differential equations whose variable coefficients are all linear functions of , because they can be transformed to the Extended Confluent Hypergeometric Equation. Consider the equation: :(A+Bz)\frac + (C+Dz)\frac +(E+Fz)w = 0 First we move the
regular singular point In mathematics, in the theory of ordinary differential equations in the complex plane \Complex, the points of \Complex are classified into ''ordinary points'', at which the equation's coefficients are analytic functions, and ''singular points'', at ...
to by using the substitution of , which converts the equation to: :z\frac + (C+Dz)\frac +(E+Fz)w = 0 with new values of , and . Next we use the substitution: : z \mapsto \frac z and multiply the equation by the same factor, obtaining: :z\frac+\left(C+\fracz\right)\frac+\left(\frac+\fracz\right)w=0 whose solution is :\exp \left ( - \left (1+ \frac \right) \frac \right )w(z), where is a solution to Kummer's equation with :a=\left (1+ \frac \right)\frac-\frac, \qquad b = C. Note that the square root may give an imaginary or complex number. If it is zero, another solution must be used, namely :\exp \left(-\tfrac Dz \right )w(z), where is a
confluent hypergeometric limit function In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, wh ...
satisfying :zw''(z)+Cw'(z)+\left(E-\tfracCD \right)w(z)=0. As noted below, even the
Bessel equation Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
can be solved using confluent hypergeometric functions.


Integral representations

If , can be represented as an integral :M(a,b,z)= \frac\int_0^1 e^u^(1-u)^\,du. thus is the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
of the
beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
. For with positive real part can be obtained by the Laplace integral :U(a,b,z) = \frac\int_0^\infty e^t^(1+t)^\,dt, \quad (\operatorname\ a>0) The integral defines a solution in the right half-plane . They can also be represented as
Barnes integral In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by . They are closely related to generalized hypergeometric series. The integral is usually taken a ...
s :M(a,b,z) = \frac\frac\int_^ \frac(-z)^sds where the contour passes to one side of the poles of and to the other side of the poles of .


Asymptotic behavior

If a solution to Kummer's equation is asymptotic to a power of as , then the power must be . This is in fact the case for Tricomi's solution . Its
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
behavior as can be deduced from the integral representations. If , then making a change of variables in the integral followed by expanding the
binomial series In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like (1+x)^n for a nonnegative integer n. Specifically, the binomial series is the Taylor series for the function f(x)=(1+x) ...
and integrating it formally term by term gives rise to an
asymptotic series In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
expansion, valid as : :U(a,b,x)\sim x^ \, _2F_0\left(a,a-b+1;\, ;-\frac 1 x\right), where _2F_0(\cdot, \cdot; ;-1/x) is a
generalized hypergeometric series In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...
with 1 as leading term, which generally converges nowhere, but exists as a
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
in . This
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
is also valid for complex instead of real , with The asymptotic behavior of Kummer's solution for large is: :M(a,b,z)\sim\Gamma(b)\left(\frac+\frac\right) The powers of are taken using . The first term is not needed when is finite, that is when is not a non-positive integer and the real part of goes to negative infinity, whereas the second term is not needed when is finite, that is, when is a not a non-positive integer and the real part of goes to positive infinity. There is always some solution to Kummer's equation asymptotic to as . Usually this will be a combination of both and but can also be expressed as .


Relations

There are many relations between Kummer functions for various arguments and their derivatives. This section gives a few typical examples.


Contiguous relations

Given , the four functions are called contiguous to . The function can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of , and . This gives relations, given by identifying any two lines on the right hand side of :\begin z\frac = z\fracM(a+,b+) &=a(M(a+)-M)\\ &=(b-1)(M(b-)-M)\\ &=(b-a)M(a-)+(a-b+z)M\\ &=z(a-b)M(b+)/b +zM\\ \end In the notation above, , , and so on. Repeatedly applying these relations gives a linear relation between any three functions of the form (and their higher derivatives), where , are integers. There are similar relations for .


Kummer's transformation

Kummer's functions are also related by Kummer's transformations: :M(a,b,z) = e^z\,M(b-a,b,-z) :U(a,b,z)=z^ U\left(1+a-b,2-b,z\right).


Multiplication theorem

The following
multiplication theorem Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additio ...
s hold true: :\begin U(a,b,z) &= e^ \sum_ \frac U(a,b+i,z t)\\ &= e^ t^ \sum_ \frac U(a-i,b-i,z t). \end


Connection with Laguerre polynomials and similar representations

In terms of
Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions on ...
, Kummer's functions have several expansions, for example :M\left(a,b,\frac\right) = (1-x)^a \cdot \sum_n\fracL_n^(y)x^n


Special cases

Functions that can be expressed as special cases of the confluent hypergeometric function include: *Some
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
s where the left-hand side is not defined when is a non-positive integer, but the right-hand side is still a solution of the corresponding Kummer equation: ::M(0,b,z)=1 ::U(0,c,z)=1 ::M(b,b,z)=e^z ::U(a,a,z)=e^z\int_z^\infty u^e^du (a polynomial if is a non-positive integer) ::\frac+\frac=z^e^z ::M(n,b,z) for non-positive integer is a
generalized Laguerre polynomial In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only ...
. ::U(n,c,z) for non-positive integer is a multiple of a generalized Laguerre polynomial, equal to \tfracM(n,c,z) when the latter exists. ::U(c-n,c,z) when is a positive integer is a closed form with powers of , equal to \tfracz^M(1-n,2-c,z) when the latter exists. ::U(a,a+1,z)= z^ ::U(-n,-2n,z) for non-negative integer is a Bessel polynomial (see lower down). ::M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2 etc. ::Using the contiguous relation aM(a+)=(a+z)M+z(a-b)M(b+)/b we get, for example, M(2,1,z)=(1+z)e^z. * Bateman's function *
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s and many related functions such as
Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solutio ...
s,
Kelvin function In applied mathematics, the Kelvin functions ber''ν''(''x'') and bei''ν''(''x'') are the real and imaginary parts, respectively, of :J_\nu \left (x e^ \right ),\, where ''x'' is real, and , is the ''ν''th order Bessel function of the first kin ...
s,
Hankel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s. For example, in the special case the function reduces to a
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
: ::_1F_1(a,2a,x)= e^\, _0F_1 \left(; a+\tfrac; \tfrac \right) = e^ \left(\tfrac\right)^\Gamma\left(a+\tfrac\right)I_\left(\tfrac\right). :This identity is sometimes also referred to as Kummer's second transformation. Similarly ::U(a,2a,x)= \frac x^ K_ (x/2), :When is a non-positive integer, this equals where is a
Bessel polynomial In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series :y_n(x)=\sum_^n\frac\,\left(\frac ...
. * The
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non-elementary ...
can be expressed as ::\mathrm(x)= \frac\int_0^x e^ dt= \frac\ _1F_1\left(\tfrac,\tfrac,-x^2\right). *
Coulomb wave function In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of conflu ...
* Cunningham functions *
Exponential integral In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. Definitions For real non-zero values of&n ...
and related functions such as the
sine integral In mathematics, trigonometric integrals are a indexed family, family of integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operatorname(x) = -\int ...
,
logarithmic integral In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
* Hermite polynomials *
Incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which ...
*
Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions on ...
*
Parabolic cylinder function In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabo ...
(or Weber function) *
Poisson–Charlier function In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier. They are given in terms of the generalized hypergeometric function by :C_n(x; \mu)= _2F_0(-n,-x;-; ...
*
Toronto function In mathematics, the Toronto function ''T''(''m'',''n'',''r'') is a modification of the confluent hypergeometric function defined by , Weisstein, as :T(m,n,r)=r^e^\frac_1F_1(m+;n+1;r^2). :Later, Heatley (1964) recomputed to 12 decimals the table of ...
s *
Whittaker function In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced W ...
s are solutions of
Whittaker's equation In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced Wh ...
that can be expressed in terms of Kummer functions and by ::M_(z) = e^ z^M\left(\mu-\kappa+\tfrac, 1+2\mu; z\right) ::W_(z) = e^ z^U\left(\mu-\kappa+\tfrac, 1+2\mu; z\right) * The general -th raw moment ( not necessarily an integer) can be expressed as :: \begin \operatorname \left N\left(\mu, \sigma^2 \right)\^p \right&= \frac \ _1F_1\left(-\tfrac p 2, \tfrac 1 2, -\tfrac\right)\\ \operatorname \left \left(\mu, \sigma^2 \right)^p \right&= \left (-2 \sigma^2\right)^ U\left(-\tfrac p 2, \tfrac 1 2, -\tfrac \right) \end :In the second formula the function's second
branch cut In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
can be chosen by multiplying with .


Application to continued fractions

By applying a limiting argument to
Gauss's continued fraction In complex analysis, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several i ...
it can be shown that :\frac = \cfrac and that this continued fraction converges uniformly to a
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
of in every bounded domain that does not include a pole.


Notes


References

* * * * * * * * * {{cite book , last1=Oldham , first1=K.B. , last2=Myland , first2=J. , last3=Spanier , first3=J. , title=An Atlas of Functions: with Equator, the Atlas Function Calculator , publisher=Springer New York , series=An Atlas of Functions , year=2010 , isbn=978-0-387-48807-3 , url=https://books.google.com/books?id=UrSnNeJW10YC&pg=PA75 , access-date=2017-08-23


External links


Confluent Hypergeometric Functions
in NIST Digital Library of Mathematical Functions
Kummer hypergeometric function
on the Wolfram Functions site
Tricomi hypergeometric function
on the Wolfram Functions site Hypergeometric functions Special hypergeometric functions Special functions