Appell Functions

Appell Functions#

Appell function \(F_1\)#

ctx.appell_f1(a, b1, b2, c, x, y)#

where ctx is dec, mpm, fpm, or gmp.

Returns the Appell function F1(). See also Wikipedia [1482], MathWorld [1114], MathWorld [1122], NIST [26], Mpmath [768].

Gives the Appell F1 hypergeometric function of two variables,

\[F_1(a,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c)_{m+n}} \frac{x^m y^n}{m! n!}.\]

This series is only generally convergent when \(|x| < 1\) and \(|y| < 1\), although the function can evaluate an analytic continuation with respecto to either variable, and sometimes both.

Examples

Evaluation is supported for real and complex parameters:

>>> from mpfunlab import *
>>> mp.dps = 25; mp.pretty = True
>>> appellf1(1,0,0.5,1,0.5,0.25)
1.154700538379251529018298
>>> appellf1(1,1+j,0.5,1,0.5,0.5j)
(1.138403860350148085179415 + 1.510544741058517621110615j)

Appell function \(F_2\)#

ctx.appell_f2(a, b1, b2, c1, c2, x, y)#

where ctx is dec, mpm, fpm, or gmp.

Returns the Appell function F2().See also Wikipedia [1482], MathWorld [1114], MathWorld [1122], NIST [26], Mpmath [769].

Gives the Appell F2 hypergeometric function of two variables

\[F_2(a,b_1,b_2,c_1,c_2,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c_1)_m (c_2)_n} \frac{x^m y^n}{m! n!}.\]

The series is generally absolutely convergent for \(|x| + |y| < 1\).

Examples

Evaluation for real and complex arguments:

>>> from mpfunlab import *
>>> mp.dps = 25; mp.pretty = True
>>> appellf2(1,2,3,4,5,0.25,0.125)
1.257417193533135344785602
>>> appellf2(1,-3,-4,2,3,2,3)
-42.8
>>> appellf2(0.5,0.25,-0.25,2,3,0.25j,0.25)
(0.9880539519421899867041719 + 0.01497616165031102661476978j)
>>> chop(appellf2(1,1+j,1-j,3j,-3j,0.25,0.25))
1.201311219287411337955192
>>> appellf2(1,1,1,4,6,0.125,16)
(-0.09455532250274744282125152 - 0.7647282253046207836769297j)

Appell function \(F_3\)#

ctx.appell_f3(a1, a2, b1, b2, c, x, y)#

where ctx is dec, mpm, fpm, or gmp.

Returns the Appell function F3().See also Wikipedia [1482], MathWorld [1114], MathWorld [1122], NIST [26], Mpmath [770].

Gives the Appell F3 hypergeometric function of two variables

\[F_3(a_1,a_2,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(a_1)_m (a_2)_n (b_1)_m (b_2)_n}{(c)_{m+n}} \frac{x^m y^n}{m! n!}.\]

The series is generally absolutely convergent for \(|x| < 1, |y| < 1\).

Examples

Evaluation for various parameters and variables:

>>> from mpfunlab import *
>>> mp.dps = 25; mp.pretty = True
>>> appellf3(1,2,3,4,5,0.5,0.25)
2.221557778107438938158705
>>> appellf3(1,2,3,4,5,6,0); hyp2f1(1,3,5,6)
(-0.5189554589089861284537389 - 0.1454441043328607980769742j)
(-0.5189554589089861284537389 - 0.1454441043328607980769742j)
>>> appellf3(1,-2,-3,1,1,4,6)
-17.4
>>> appellf3(1,2,-3,1,1,4,6)
(17.7876136773677356641825 + 19.54768762233649126154534j)
>>> appellf3(1,2,-3,1,1,6,4)
(85.02054175067929402953645 + 148.4402528821177305173599j)
>>> chop(appellf3(1+j,2,1-j,2,3,0.25,0.25))
1.719992169545200286696007

Appell function \(F_4\)#

ctx.appell_f4(a, b, c1, c2, x, y)#

where ctx is dec, mpm, fpm, or gmp.

Returns the Appell function F4().See also Wikipedia [1482], MathWorld [1114], MathWorld [1122], NIST [26], Mpmath [771].

Gives the Appell F4 hypergeometric function of two variables

\[F_4(a,b,c_1,c_2,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(a)_{m+n} (b)_{m+n}}{(c_1)_m (c_2)_n} \frac{x^m y^n}{m! n!}.\]

The series is generally absolutely convergent for \(\sqrt{|x|} + \sqrt{|y|} < 1\).

Examples

Evaluation for various parameters and arguments:

>>> from mpfunlab import *
>>> mp.dps = 25; mp.pretty = True
>>> appellf4(1,1,2,2,0.25,0.125)
1.286182069079718313546608
>>> appellf4(-2,-3,4,5,4,5)
34.8
>>> appellf4(5,4,2,3,0.25j,-0.125j)
(-0.2585967215437846642163352 + 2.436102233553582711818743j)
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