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| In [[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. (The term "[[Confluence|confluent]]" 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:
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| *'''Kummer's (confluent hypergeometric) function''' ''M''(''a'',''b'',''z''), introduced by {{harvs|txt|authorlink=Ernst Kummer|last=Kummer|year=1837}}, is a solution to '''Kummer's differential equation'''. There is a different and unrelated [[Kummer's function]] bearing the same name.
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| *'''Tricomi's (confluent hypergeometric) function''' ''U''(''a'';''b'';''z'') introduced by {{harvs|txt|authorlink=Francesco Tricomi|first=Francesco|last=Tricomi|year=1947}}, sometimes denoted by Ψ(''a'';''b'';''.z''), is another solution to Kummer's equation.
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| * '''[[Whittaker function]]s''' (for [[Edmund Taylor Whittaker]]) are solutions to '''Whittaker's equation'''.
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| *'''[[Coulomb wave function]]s''' are solutions to the '''Coulomb wave equation'''.
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| 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.
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| ==Kummer's equation==
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| Kummer's equation is
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| :<math>z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0</math>,
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| with a regular singular point at 0 and an irregular singular point at ∞. | |
| It has two (usually) [[linearly independent]] solutions ''M''(''a'',''b'',''z'') and ''U''(''a'',''b'',''z'').
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| Kummer's function (of the first kind) ''M'' is a [[generalized hypergeometric series]] introduced in {{harv|Kummer|1837}}, given by
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| :<math>M(a,b,z)=\sum_{n=0}^\infty \frac {a^{(n)} z^n} {b^{(n)} n!}={}_1F_1(a;b;z)</math>
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| where
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| : <math>a^{(0)}=1,</math>
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| : <math>a^{(n)}=a(a+1)(a+2)\cdots(a+n-1)\,</math>
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| is the [[rising factorial]]. Another common notation for this solution is Φ(''a'',''b'',''z''). Considered as a function of ''a'', ''b'', or ''z'' with the other two held constant, this defines an [[entire function]] of ''a'' or ''z'', except when ''b'' = 0, −1, − 2, ... As a function of ''b'' it is [[analytic function|analytic]] except for poles at the non-positive integers.
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| Some values of ''a'' and ''b'' yield solutions that can be expressed in terms of other known functions. See [[#Special cases]]. When ''a'' is a non-positive integer then Kummer's function (if it is defined) is a (generalized) [[Laguerre polynomial]].
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| 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]]
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| :<math>M(a,c,z) = \lim_{b\rightarrow\infty}{}_2F_1(a,b;c;z/b)</math>
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| and many of the properties of the confluent hypergeometric function are limiting cases of properties of the hypergeometric function.
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| Since Kummer's equation is second order there must be another, independent, solution. For this we can usually use the Tricomi confluent hypergeometric function ''U''(''a'';''b'';''z'') introduced by {{harvs|txt|authorlink=Francesco Tricomi|first=Francesco|last=Tricomi|year=1947}}, and sometimes denoted by Ψ(''a'';''b'';''.z'').
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| The function ''U'' is defined in terms of Kummer's function ''M'' by
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| :<math>U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a-b+1)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a-b+1,2-b,z).</math>
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| This is undefined for integer ''b'', but can be extended to integer ''b'' by continuity. Unlike Kummer's function which is an [[entire function]] of ''z'', ''U''(''z'') usually has a [[singularity]] at zero. But see [[#Special cases]] for some examples where it is an entire function (polynomial).
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| Note that if <math>\Gamma(b-1)/\Gamma(a)</math> is zero (which can occur if ''a'' is a non-positive integer), then <math>U(a,b,z)</math> and <math>M(a,b,z)</math> are not independent and another solution is needed. Also when ''b'' is a non-positive integer we need another solution because then <math>M(a,b,z)</math> is not defined. For instance, if ''a'' = 0 and ''b'' = 0, Kummer's function is undefined, but two independent solutions are <math>w(z)=U(0,0,z)=1</math> and <math>w(z)=\exp(z).</math> For ''a'' = 0 but at other values of ''b'', we have the two solutions:
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| :<math>U(0,b,z)=1</math>
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| :<math>w(z)=\int_{-\infty}^zu^{-b}e^u\mathrm{d}u</math>
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| When ''b'' = 1 this second solution is the [[exponential integral]] Ei(''z'').
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| See [[#Special cases]] for solutions to some other cases.
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| ===Other equations===
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| Confluent hypergeometric functions can be used to solve "most" second-order differential equations in which the coefficients are all linear functions of ''z'':
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| :<math>(A+Bz)\frac{d^2w}{dz^2} + (C+Dz)\frac{dw}{dz} +(E+Fz)w = 0</math>
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| First of all, a substitution of ''A''+''Bz'' with a new ''z'' converts the equation to:
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| :<math>z\frac{d^2w}{dz^2} + (C+Dz)\frac{dw}{dz} +(E+Fz)w = 0</math>
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| with new values of ''C'', ''D'', ''E'', and ''F''. (This step simply moves the [[regular singular point]] to 0.) If we then replace this z with <math>1/\sqrt{D^2-4F}</math> times a new ''z'', and multiply the equation by the same factor, we get:
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| :<math>z\frac{d^2w}{dz^2}+\left(C+\frac D\sqrt{D^2-4F}z\right)\frac{dw}{dz}+\left(\frac E\sqrt{D^2-4F}+\frac F{D^2-4F}z\right)w=0</math>
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| whose solution is <math>\exp[-(1+D/(D^2-4F)^{1/2})z/2]w(z)</math>, where ''w''(''z'') is a solution to Kummer's equation with <math>a=(1+D/(D^2-4F)^{1/2})C/2-E/\sqrt{D^2-4F}</math> and <math>b=C</math>. Note that the square root may give an imaginary (or complex) number. If it is zero, another solution must be used, namely <math>\exp(-Dz/2)w(z)</math>, where ''w''(''z'') is a [[confluent hypergeometric limit function]] satisfying <math>zw''(z)+Cw'(z)+(E-CD/2)w(z)=0.</math>
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| As noted lower down, even the [[Bessel equation]] can be solved using confluent hypergeometric functions.
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| ==Integral representations==
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| If Re ''b'' > Re ''a'' > 0, ''M''(''a'',''b'',''z'') can be represented as an integral
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| :<math>M(a,b,z)= \frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}\int_0^1 e^{zu}u^{a-1}(1-u)^{b-a-1}\,du\,\quad .</math>
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| thus <math>M(a,a+b,it)</math> is the [[characteristic function (probability)|characteristic function]] of the [[beta distribution]]. For ''a'' with positive real part ''U'' can be obtained by the [[Laplace transform|Laplace integral]]
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| :<math>U(a,b,z) = \frac{1}{\Gamma(a)}\int_0^\infty e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt, \quad (\operatorname{re}\ a>0) </math>
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| The integral defines a solution in the right half-plane re ''z'' > 0.
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| They can also be represented as [[Barnes integral]]s
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| :<math>M(a,b,z) = \frac{1}{2\pi i}\frac{\Gamma(b)}{\Gamma(a)}\int_{-i\infty}^{i\infty} \frac{\Gamma(-s)\Gamma(a+s)}{\Gamma(b+s)}(-z)^sds</math>
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| where the contour passes to one side of the poles of Γ(−''s'') and to the other side of the poles of Γ(''a''+''s'').
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| ==Asymptotic behavior== | |
| If a solution to Kummer's equation is asymptotic to a power of ''z'' as ''z'' goes to infinity, then the power must be −''a''. This is in fact the case for Tricomi's solution ''U''(''a'',''b'',''z''). Its [[asymptotic]] behavior as ''z'' → ∞ can be deduced from the integral representations.
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| If ''z'' = ''x'' is real, then making a change of variables in the integral followed by expanding the [[binomial series]] and integrating it formally term by term gives rise to an [[asymptotic series]] expansion, valid as ''x'' → ∞:<ref>{{Cite book|title=Special functions|last1=Andrews|first1=G.E.|last2=Askey|first2=R.|last3=Roy|first3=R.|year=2001|publisher=Cambridge University Press|isbn=978-0521789882}}.
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| </ref>
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| :<math>U(a,b,x)\sim x^{-a} \, _2F_0\left(a,a-b+1;\, ;-\frac 1 x\right),</math>
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| where <math>_2F_0(\cdot, \cdot; ;-1/x)</math> is a [[generalized hypergeometric series]] (with 1 as leading term), which generally converges nowhere but exists as a [[formal power series]] in 1/''x''. This [[asymptotic expansion]] is also valid for complex ''z'' instead of real ''x'', with <math>|\arg z|<\tfrac 3 2 \pi.</math>
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| The asymptotic behavior of Kummer's solution for large |''z''| is: | |
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| :<math>M(a,b,z)\sim\Gamma(b)\left(\frac{e^zz^{a-b}}{\Gamma(a)}+\frac{e^{-i\pi a}z^{-a}}{\Gamma(b-a)}\right)</math>
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| The powers of ''z'' are taken using <math>-\tfrac 3 2\pi<\arg z\le\tfrac 1 2\pi</math>.<ref>This is derived from Abramowitz and Stegun (see reference below), [http://people.math.sfu.ca/~cbm/aands/page_508.htm page 508]. They give a full asymptotic series. They switch the sign of the exponent in exp(''iπa'') in the right half-plane but this is unimportant because the term is negligible there or else ''a'' is an integer and the sign doesn't matter.</ref> The first term is only needed when Γ(''b-a'') is infinite (that is, when ''b-a'' is a non-positive integer) or when the real part of ''z'' is non-negative, whereas the second term is only needed when Γ(''a'') is infinite (that is, when ''a'' is a non-positive integer) or when the real part of ''z'' is non-positive.
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| There is always some solution to Kummer's equation asymptotic to <math>e^zz^{a-b}</math> as ''z'' goes to minus infinity. Usually this will be a combination of both <math>M(a,b,z)\text{ and }U(a,b,z)</math> but can also be expressed as <math>e^z(-1)^{a-b}U(b-a,b,-z)</math>.
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| ==Relations==
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| There are many relations between Kummer functions for various arguments and their derivatives. This section gives a few typical examples.
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| ===Contiguous relations===
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| Given ''M''(''a'', ''b''; ''z''), the four functions ''M''(''a'' ± 1, ''b'', ''z''), ''M''(''a'', ''b'' ± 1; ''z'') are called contiguous to ''M''(''a'', ''b''; ''z''). The function ''M''(''a'', ''b''; ''z'') can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of ''a'', ''b'', and ''z''. This gives ({{su|p=4|b=2}})=6 relations, given by identifying any two lines on the right hand side of
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| :<math>\begin{align} | |
| z\frac{dM}{dz} = z\frac{a}{b}M(a+,b+)
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| &=a(M(a+)-M)\\
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| &=(b-1)(M(b-)-M)\\
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| &=(b-a)M(a-)+(a-b+z)M\\
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| &=z(a-b)M(b+)/b +zM\\
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| \end{align}</math>
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| In the notation above, ''M'' = ''M''(''a'', ''b''; ''z''), ''M''(''a''+) = ''M''(''a'' + 1, ''b''; ''z''), and so on.
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| Repeatedly applying these relations gives a linear relation between any three functions of the form ''M''(''a'' + ''m'', ''b'' + ''n''; ''z'') (and their higher derivatives), where ''m'', ''n'' are integers.
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| There are similar relations for ''U''.
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| ===Kummer's transformation===
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| Kummer's functions are also related by Kummer's transformations:
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| :<math>M(a,b,z) = e^z\,M(b-a,b,-z)</math>
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| :<math>U(a,b,z)=z^{1-b} U\left(1+a-b,2-b,z\right)</math>.
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| ==Multiplication theorem==
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| The following [[multiplication theorem]]s hold true:
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| :<math>\begin{align}U(a,b,z)&= e^{(1-t)z} \sum_{i=0} \frac{(t-1)^i z^i}{i!} U(a,b+i,z t)=\\
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| &= e^{(1-t)z} t^{b-1} \sum_{i=0} \frac{\left(1-\frac 1 t\right)^i}{i!} U(a-i,b-i,z t).\end{align}</math>
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| ==Connection with Laguerre polynomials and similar representations==
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| In terms of [[Laguerre polynomials]], Kummer's functions have several expansions, for example
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| :<math>M\left(a,b,\frac{x y}{x-1}\right) = (1-x)^a \cdot \sum_n\frac{a^{(n)}}{b^{(n)}}L_n^{(b-1)}(y)x^n</math> {{harv|Erdelyi|1953|loc=6.12}}
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| ==Special cases==
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| Functions that can be expressed as special cases of the confluent hypergeometric function include:
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| *Some [[elementary function]]s (the left-hand side is not defined when ''b'' is a non-positive integer, but the right-hand side is still a solution of the corresponding Kummer equation):
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| :<math>M(0,b,z)=1</math>
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| :<math>U(0,c,z)=1</math>
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| :<math>M(b,b,z)=\exp(z)</math>
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| :<math>U(a,a,z)=\exp(z)\int_z^\infty u^{-a}\exp(-u)du</math> (a polynomial if ''a'' is a non-positive integer)
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| :<math>\frac{U(1,b,z)}{\Gamma(b-1)}+\frac{M(1,b,z)}{\Gamma(b)}=z^{1-b}\exp(z)</math>
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| :<math>U(a,a+1,z)= z^{-a}\,</math>
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| :<math>U(-n,-2n,z)</math> for integer ''n'' is a Bessel polynomial (see lower down).
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| :<math>M(n,b,z)</math> for non-positive integer ''n'' is a [[generalized Laguerre polynomial]].
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| *[[Bateman's function]]
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| *[[Bessel function]]s and many related functions such as [[Airy function]]s, [[Kelvin function]]s, [[Hankel function]]s.
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| For example, the special case <math>b= 2 a</math> the function reduces to a [[Bessel function]]:
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| :<math>\begin{align}\, _1F_1(a,2a,x)&= e^{\frac x 2}\, _0F_1 (; a+\tfrac{1}{2}; \tfrac{1}{16}x^2) \\
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| &= e^{\frac x 2} \left(\tfrac{1}{4}x\right)^{\tfrac{1}{2}-a} \Gamma\left(a+\tfrac{1}{2}\right) I_{a-\frac 1 2}\left(\tfrac{1}{2}x\right).\end{align}</math>
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| This identity is sometimes also referred to as [[Ernst Kummer|Kummer's]] second transformation.
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| Similarly
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| :<math>U(a,2a,x)= \frac{e^\frac x 2}{\sqrt \pi} x^{\frac 1 2 -a} K_{a-\frac 1 2} \left(\frac x 2 \right),</math>
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| When <math>a</math> is a non-positive integer, this equals <math>2^{-a}\theta_{-a}(x/2)</math> where θ is a [[Bessel polynomial]].
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| * The [[error function]] can be expressed as
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| ::<math>\mathrm{erf}(x)= \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt=
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| \frac{2x}{\sqrt{\pi}}\,_1F_1\left(\frac{1}{2},\frac{3}{2},-x^2\right).</math>
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| *[[Coulomb wave function]]
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| *[[Cunningham function]]s
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| *[[Exponential integral]] and related functions such as the [[sine integral]], [[logarithmic integral]]
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| *[[Hermite polynomials]]
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| *[[Incomplete gamma function]]
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| *[[Laguerre polynomials]]
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| *[[Parabolic cylinder function]] (or Weber function)
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| *[[Poisson–Charlier function]]
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| *[[Toronto function]]s
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| *[[Whittaker function]]s ''M''<sub>κ,μ</sub>(''z''), ''W''<sub>κ,μ</sub>(''z'') are solutions of [[Whittaker's equation]] that can be expressed in terms of Kummer functions ''M'' and ''U'' by
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| :<math>M_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\frac{1}{2}, 1+2\mu; z\right)</math>
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| :<math>W_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\frac{1}{2}, 1+2\mu; z\right)</math>
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| * The general p-''th'' raw moment (p not necessarily an integer) can be expressed as
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| :: <math>\operatorname{E} \left[\left|N\left(\mu, \sigma^2 \right)\right|^p \right]= \left(2 \sigma^2\right)^\frac p 2 \frac {\Gamma\left(\frac{1+p}2\right)}{\sqrt \pi}\, _1F_1\left(-\frac p 2, \frac 1 2, -\frac{\mu^2}{2 \sigma^2}\right),</math>
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| :: <math>\operatorname{E} \left[N\left(\mu, \sigma^2 \right)^p \right]=(-2 \sigma^2)^\frac p 2 \cdot U\left(-\frac p 2, \frac 1 2, -\frac{\mu^2}{2 \sigma^2} \right)</math> (the function's second [[branch cut]] can be chosen by multiplying with <math>(-1)^p</math>).
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| ==Application to continued fractions==
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| By applying a limiting argument to [[Gauss's continued fraction]] it can be shown that
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| :<math>
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| \frac{M(a+1,b+1,z)}{M(a,b,z)} = \cfrac{1}{1 - \cfrac{{\displaystyle\frac{b-a}{b(b+1)}z}}
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| {1 + \cfrac{{\displaystyle\frac{a+1}{(b+1)(b+2)}z}}
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| {1 - \cfrac{{\displaystyle\frac{b-a+1}{(b+2)(b+3)}z}}
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| {1 + \cfrac{{\displaystyle\frac{a+2}{(b+3)(b+4)}z}}{1 - \ddots}}}}}
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| </math>
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| and that this continued fraction converges uniformly to a meromorphic function of ''z'' in every bounded domain that does not include a pole.
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| ==Notes==
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| <references/>
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| ==References==
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| * {{AS ref |13|504}}
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| * {{springer|id=c/c024700|first=E.A. |last=Chistova}}
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| * {{dlmf|first=Adri B. Olde|last= Daalhuis|id=13}}
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| * {{cite book | last1= Erdélyi | first1= Arthur | author1-link= Arthur Erdélyi | last2= Magnus | first2= Wilhelm | author2-link= Wilhelm Magnus | last3= Oberhettinger | first3= Fritz | lastauthoramp= yes | last4= Tricomi | first4= Francesco G. | title= Higher transcendental functions. Vol. I | location= New York–Toronto–London | publisher= McGraw–Hill Book Company, Inc. | year= 1953 | mr= 0058756 | ref= harv}}
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| * {{cite journal | last= Kummer | first= Ernst Eduard | authorlink= Ernst Eduard Kummer | title= De integralibus quibusdam definitis et seriebus infinitis | language= Latin | url= http://resolver.sub.uni-goettingen.de/purl?GDZPPN002141329 | format= | journal= [[Journal für die reine und angewandte Mathematik]] | year= 1837 | volume= 17 | pages= 228–242 | issn= 0075-4102 | ref= harv}}
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| * {{cite book | last= Slater | first= Lucy Joan | authorlink= Lucy Joan Slater | title= Confluent hypergeometric functions | location= Cambridge, UK | publisher= Cambridge University Press | year= 1960 | mr= 0107026 | ref= harv}}
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| * {{cite journal | last= Tricomi | first= Francesco G. | authorlink= Francesco Giacomo Tricomi | title= Sulle funzioni ipergeometriche confluenti | language= Italian | journal= Annali di Matematica Pura ed Applicata. Serie Quarta | year= 1947 | volume= 26 | pages= 141–175 | issn= 0003-4622 | mr= 0029451 | ref= harv}}
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| * {{cite book | last= Tricomi | first= Francesco G. | title= Funzioni ipergeometriche confluenti | language= Italian | location= Rome | publisher= Edizioni cremonese | year= 1954 | series= Consiglio Nazionale Delle Ricerche Monografie Matematiche | volume= 1 | isbn= 978-88-7083-449-9 | mr= 0076936 | ref=harv}}
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| ==External links==
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| * [http://dlmf.nist.gov/13 Confluent Hypergeometric Functions] in NIST Digital Library of Mathematical Functions
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| * [http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/ Kummer hypergeometric function] on the Wolfram Functions site
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| * [http://functions.wolfram.com/HypergeometricFunctions/HypergeometricU/ Tricomi hypergeometric function] on the Wolfram Functions site
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| [[Category:Hypergeometric functions]]
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| [[Category:Special hypergeometric functions]]
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