Confluent hypergeometric function: Difference between revisions

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Revision as of 23:47, 26 August 2020

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:<math>z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0,</math>

with a regular singular point at <{{math>|''z'' {{=}} 0</math>}} and an irregular singular point at <{{math>|''z'' {{=\infty</math>}} ∞}}. It has two (usually) [[linearly independent]] solutions {{math|''M''(''a'', ''b'', ''z'')}} and {{math|''U''(''a'', ''b'', ''z'')}}.

Kummer's function of the first kind ''{{mvar|M''}} is a [[generalized hypergeometric series]] introduced in {{harv|Kummer|1837}}, given by:

:<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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: <math>a^{(n)}=a(a+1)(a+2)\cdots(a+n-1)\, ,</math>

is the [[rising factorial]]. Another common notation for this solution is {{math|Φ(''a'', ''b'', ''z'')}}. Considered as a function of ''{{mvar|a''}}, ''{{mvar|b''}}, or {{mvar|z}} with the other two held constant, this defines an [[entire function]] of ''{{mvar|a''}} or ''{{mvar|z''}}, except when {{math|''b'' {{=}} 0, −1, −2, ...}} As a function of {{mvar|b}} it is [[analytic function|analytic]] except for poles at the non-positive integers.

Some values of ''{{mvar|a''}} and {{mvar|b}} yield solutions that can be expressed in terms of other known functions. See [[#Special cases]]. When ''{{mvar|a''}} is a non-positive integer, then Kummer's function (if it is defined) is a generalized [[Laguerre polynomial]].

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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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 {{math|1 − ''b''}}. If we let {{math|''w''(''z'')}} be

:<math>w(z)=z^{1-b}v(z)</math>

then the differential equation gives

:<math>z^{2-b}\frac{d^2v}{dz^2}+2(1-b)z^{1-b}\frac{dv}{dz}-b(1-b)z^{-b}v + (b-z)\left[z^{1-b}\frac{dv}{dz}+(1-b)z^{-b}v\right] - az^{1-b}v = 0</math>

which, upon dividing out <{{math>|''z^{1-''<sup>1−''b}''</mathsub>}} and simplifying, becomes

<!--:<math>z\frac{d^2v}{dz^2}+2(1-b)\frac{dv}{dz}-b(1-b)z^{-1}v + (b-z)\left[\frac{dv}{dz}+(1-b)z^{-1}v\right] - av = 0</math>-->

:<math>z\frac{d^2v}{dz^2}+(2-b-z)\frac{dv}{dz} - (a+1-b)v = 0.</math>

This means that <{{math>|''z^{1-''^1−''b}''''M''(''a'' + 1- − ''b'', 2- − ''b'', ''z'')</math>}} is a solution so long as ''{{mvar|b''}} is not an integer greater than 1, just as <{{math>|''M''(''a'', ''b'', ''z'')</math>}} is a solution so long as ''{{mvar|b''}} is not an integer less than 1. We can also use the Tricomi confluent hypergeometric function {{math|''U''(''a'', ''b'', ''z'')}} introduced by {{harvs|txt|authorlink=Francesco Tricomi| first=Francesco |last=Tricomi | year=1947}}, and sometimes denoted by {{math|Ψ(''a''; ''b''; ''z'')}}. It is a combination of the above two solutions, defined by

:<math>U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a+1-b)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a+1-b,2-b,z).</math>

Although this expression is undefined for integer {{mvar|b}}, it has the advantage that it can be extended to any integer {{mvar|b}} by continuity. Unlike Kummer's function which is an [[entire function]] of ''{{mvar|z''}}, {{math|''U''(''z'')}} usually has a [[singularity (mathematics)|singularity]] at zero. For example, if {{math|''b'' {{=}} 0}} and {{math|''a''≠0 ≠ 0}} then <{{math>\Gamma|Γ(''a''+1)''U''(''a'', ''b'', ''z'')- − 1</math>}} is asymptotic to <{{math>|''az\'' ln z</math> as ''z''}} as {{mvar|z}} goes to zero. But see [[#Special cases]] for some examples where it is an entire function (polynomial).

Note that the solution <{{math>|''z^{1-''^1−''b}U''''M''(''a'' + 1- − ''b'', 2- − ''b'', ''z'')</math>}} to Kummer's equation is the same as the solution <{{math>|''U''(''a'', ''b'', ''z'')</math>}}, see [[#Kummer's transformation]].

For most combinations of real or complex ''{{mvar|a''}} and ''{{mvar|b''}}, the functions <{{math>|''M''(''a'', ''b'', ''z'')</math>}} and <{{math>|''U''(''a'', ''b'', ''z'')</math>}} are independent, and if ''{{mvar|b''}} is a non-positive integer, so <{{math>|''M''(''a'', ''b'', ''z'')</math>}} doesn't exist, then we may be able to use <{{math>|''z^{1-''<sup>1−''b}''</sub>''M''(''a''+1-1−''b'',2- 2−''b'', ''z'')</math>}} as a second solution. But if {{mvar|a}} is a non-positive integer and {{mvar|b}} is not a non-positive integer, then {{math|''U''(''z'')}} is a multiple of {{math|''M''(''z'')}}. In that case as well, <{{math>|''z^{1-''<sup>1−''b}''</sub>''M''(''a''+1-1−''b'',2- 2−''b'', ''z'')</math>}} can be used as a second solution if it exists and is different. But when ''{{mvar|b''}} is an integer greater than 1, this solution doesn't exist, and if {{math|1=''b'' = 1}} then it exists but is a multiple of <{{math>|''U''(''a'', ''b'', ''z'')</math>}} and of <{{math>|''M''(''a'', ''b'', ''z'').</math>}} In those cases a second solution exists of the following form and is valid for any real or complex ''{{mvar|a''}} and any positive integer ''{{mvar|b''}} except when ''{{mvar|a''}} is a positive integer less than ''{{mvar|b''}}:

:<math>M(a,b,z)\ln z+z^{1-b}\sum_{k=0}^\infty C_kz^k</math>

When ''a'' = 0 we can alternatively use:

:<math>\int_{-\infty}^z(-u)^{-b}e^u\mathrm{d}u.</math>

When <{{math>|''b'' {{=}} 1</math>}} this is the [[exponential integral]] {{math|''E''₁(''-z−z'')}}.

A similar problem occurs when {{math|''a''−''b''}} is a negative integer and ''{{mvar|b''}} is an integer less than 1. In this case <{{math>|''M''(''a'', ''b'', ''z'')</math>}} doesn't exist, and <{{math>|''U''(''a'', ''b'', ''z'')</math>}} is a multiple of <{{math>|''z^{1-''^1−''b}''''M''(''a''+1-1−''b'',2- 2−''b'', ''z'').</math>}} A second solution is then of the form:

:<math>z^{1-b}M(a+1-b,2-b,z)\ln z+\sum_{k=0}^\infty C_kz^k</math>

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:<math>z\frac{d^2w}{dz^2} +(b-z)\frac{dw}{dz} -\left(\sum_{m=0}^M a_m z^m\right)w = 0</math> <ref>{{cite journal|last1=Campos|first1=LMBC|title=On Some Solutions of the Extended Confluent Hypergeometric Differential Equation|journal=Journal of Computational and Applied Mathematics|date=2001|volume=Elsevier|doi=10.1016/s0377-0427(00)00706-8|pages=177–200|doi-access=free}}</ref>

Note that for {{math|''M'' {{=}} 0}} 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 {{mvar|z}}, because they can be transformed to the Extended Confluent Hypergeometric Equation. Consider the equation:

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==Integral representations==

If {{math|Re ''b'' > Re ''a'' > 0}}, {{math|''M''(''a'', ''b'', ''z'')}} can be represented as an integral

:<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.</math>

thus <{{math>|''M''(''a'', ''a''+''b'', ''it'')</math>}} is the [[characteristic function (probability)|characteristic function]] of the [[beta distribution]]. For ''{{mvar|a''}} with positive real part {{mvar|U}} can be obtained by the [[Laplace transform|Laplace integral]]

:<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>

The integral defines a solution in the right half-plane <{{math>|0 0< \operatorname{Re} ''z'' < \pi''π''/2 </math> }}.

They can also be represented as [[Barnes integral]]s

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:<math>U(a,b,x)\sim x^{-a} \, _2F_0\left(a,a-b+1;\, ;-\frac 1 x\right),</math>

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 {{math|1/''x''}}. This [[asymptotic expansion]] is also valid for complex {{mvar|z}} instead of real ''{{mvar|x''}}, with <{{math>|\{{mabs|arg ''z|''}} <\tfrac 3 ''π''/2 \pi.</math>}}

The asymptotic behavior of Kummer's solution for large {{math|{{mabs|''z''|}}}} is:

:<math>M(a,b,z)\sim\Gamma(b)\left(\frac{e^zz^{a-b}}{\Gamma(a)}+\frac{(-z)^{-a}}{\Gamma(b-a)}\right)</math>

The powers of {{mvar|z}} are taken using <{{math>-\tfrac 3 |−3''π''/2\pi <\ arg ''z\le\tfrac'' 1≤ 2\pi<''π''/math>2}}.<ref> This is derived from Abramowitz and Stegun (see reference below), [http://people.math.sfu.ca/~cbm/aands/page_508.htm page 508], where a full asymptotic series is given. They switch the sign of the exponent in {{math|exp(''iπa'')}} in the right half-plane but this is immaterial, as the term is negligible there or else ''{{mvar|a''}} is an integer and the sign doesn't matter.</ref> The first term is not needed when {{math|Γ(''b'' − ''a'')}} is finite, that is when {{math|''b'' − ''a''}} is not a non-positive integer and the real part of {{mvar|z}} goes to negative infinity, whereas the second term is not needed when {{math|Γ(''a'')}} is finite, that is, when ''{{mvar|a''}} is a not a non-positive integer and the real part of {{mvar|z}} goes to positive infinity.

There is always some solution to Kummer's equation asymptotic to <{{math>|''e^zz''^zz^{<sup>''a-''−''b}''</mathsup>}} as {{math|''z'' → −∞}}. Usually this will be a combination of both {{math|''M''(''a'', ''b'', ''z'')}} and {{math|''U''(''a'', ''b'', ''z'')}} but can also be expressed as <{{math>|''e^^z'' (-1−1)^{^''a''-''b}'' ''U''(''b-'' − ''a'', ''b'',- −''z'')</math>}}.

==Relations==

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===Contiguous relations===

Given {{math|''M''(''a'', ''b'', ''z'')}}, the four functions {{math|''M''(''a'' ± 1, ''b'', ''z''), ''M''(''a'', ''b'' ± 1, ''z'')}} are called contiguous to {{math|''M''(''a'', ''b'', ''z'')}}. The function {{math|''M''(''a'', ''b'', ''z'')}} can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of {{mvar|a, b}}, and {{mvar|z}}. This gives {{math|1= ({{su|lh=0.8em|p=4|b=2}}) = 6}} relations, given by identifying any two lines on the right hand side of

:<math>\begin{align}

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\end{align}</math>

In the notation above, {{math|1=''M'' = ''M''(''a'', ''b'', ''z'')}}, {{math|1= ''M''(''a''+) = ''M''(''a'' + 1, ''b'', ''z'')}}, and so on.

Repeatedly applying these relations gives a linear relation between any three functions of the form {{math|''M''(''a'' + ''m'', ''b'' + ''n'', ''z'')}} (and their higher derivatives), where ''{{mvar|m''}}, ''{{mvar|n''}} are integers.

There are similar relations for ''{{mvar|U''}}.

===Kummer's transformation===

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::<math>U(0,c,z)=1</math>

::<math>M(b,b,z)=e^z</math>

::<math>U(a,a,z)=e^z\int_z^\infty u^{-a}e^{-u}du</math> (a polynomial if ''{{mvar|a''}} is a non-positive integer)

::<math>\frac{U(1,b,z)}{\Gamma(b-1)}+\frac{M(1,b,z)}{\Gamma(b)}=z^{1-b}e^z</math>

::<math>M(n,b,z)</math> for non-positive integer ''{{mvar|n''}} is a [[generalized Laguerre polynomial]].

::<math>U(n,c,z)</math> for non-positive integer ''{{mvar|n''}} is a multiple of a generalized Laguerre polynomial, equal to <math>\fractfrac{\Gamma(1-c)}{\Gamma(n+1-c)}M(n,c,z)</math> when the latter exists.

::<math>U(c-n,c,z)</math> when ''{{mvar|n''}} is a positive integer is a closed form with powers of ''{{mvar|z''}}, equal to <math>\fractfrac{\Gamma(c-1)}{\Gamma(c-n)}z^{1-c}M(1-n,2-c,z)</math> when the latter exists.

::<math>U(a,a+1,z)= z^{-a}</math>

::<math>U(-n,-2n,z)</math> for non-negative integer ''{{mvar|n''}} is a Bessel polynomial (see lower down).

::<math>M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2</math> etc.

::Using the contiguous relation <math>aM(a+)=(a+z)M+z(a-b)M(b+)/b</math> we get, for example, <math>M(2,1,z)=(1+z)e^z.</math>

*[[Bateman's function]]

*[[Bessel function]]s and many related functions such as [[Airy function]]s, [[Kelvin function]]s, [[Hankel function]]s. For example, in the special case {{math|''b'' {{=}} 2''a''}} the function reduces to a [[Bessel function]]:

::<math>{}_1F_1(a,2a,x)= e^{\frac x /2}\, {}_0F_1 \left(; a+\tfrac{1}{2}; \tfrac{x^2}{16} \right) = e^{\frac{x}{/2}} \left(\tfrac{x}{4}\right)^{\tfrac{1}{/2}-a}\Gamma\left(a+\tfrac{1}{2}\right)I_{a-\frac{1}{/2}}\left(\tfrac{x}{2}\right).</math>

:This identity is sometimes also referred to as [[Ernst Kummer|Kummer's]] second transformation. Similarly

::<math>U(a,2a,x)= \frac{e^\frac {x /2}}{\sqrt \pi} x^{\tfrac 1 /2 -a} K_{a-\tfrac 1 /2} \left(\tfrac x /2 \right),</math>

:When {{mvar|a}} is a non-positive integer, this equals <{{math>|2^{-^{−''a}\theta_{-''}''θ''_{−''a}\left''}(\tfrac ''x ''/2 \right)</math>}} where {{mvar|θ}} is a [[Bessel polynomial]].

* The [[error function]] can be expressed as

::<math>\mathrm{erf}(x)= \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt= \frac{2x}{\sqrt{\pi}}\ {}_1F_1\left(\tfrac{1}{2},\tfrac{3}{2},-x^2\right).</math>

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*[[Poisson–Charlier function]]

*[[Toronto function]]s

*[[Whittaker function]]s {{math|''M_κ,μ''(''z''), ''W_κ,μ''(''z'')}} are solutions of [[Whittaker's equation]] that can be expressed in terms of Kummer functions ''{{mvar|M''}} and ''{{mvar|U''}} by

::<math>M_{\kappa,\mu}(z) = e^{-\tfrac{z}{2}} z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu; z\right)</math>

::<math>W_{\kappa,\mu}(z) = e^{-\tfrac{z}{2}} z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu; z\right)</math>

* The general {{mvar|p}}-th raw moment ({{mvar|p}} not necessarily an integer) can be expressed as{{Citation needed|date=March 2017}}

:: <math>\begin{align}

\operatorname{E} \left[\left|N\left(\mu, \sigma^2 \right)\right|^p \right] &= \frac{\left(2 \sigma^2\right)^{\frac{p}{/2}} \Gamma\left(\tfrac{1+p}{2}\right)}{\sqrt \pi} \ {}_1F_1\left(-\tfrac p 2, \tfrac 1 2, -\tfrac{\mu^2}{2 \sigma^2}\right)\\

\operatorname{E} \left[N \left(\mu, \sigma^2 \right)^p \right] &= \left (-2 \sigma^2\right)^\frac {p /2} U\left(-\tfrac p 2, \tfrac 1 2, -\tfrac{\mu^2}{2 \sigma^2} \right)

\end{align}</math>

:In the second formula the function's second [[branch cut]] can be chosen by multiplying with <{{math>|(-1−1)^<sup>''p''</mathsup>}}.

==Application to continued fractions==