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The Hypergeometric Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapters 13 and A&S 15.
Maxima has very limited knowledge of these functions. They
can be returned from function hgfred
.
Whittaker M function (A&S eqn 13.1.32):
where \(M(a,b,z)\) is Kummer’s solution of the confluent hypergeometric equation.
This can also be expressed by the series (DLMF 13.14.E6):
Whittaker W function (A&S eqn 13.1.33):
where \(U(a,b,z)\) is Kummer’s second solution of the confluent hypergeometric equation.
The \(_{p}F_{q}(a_1,a_2,...,a_p;b_1,b_2,...,b_q;z)\) hypergeometric function, where a a list of length p and b a list of length q.
The hypergeometric function. Unlike Maxima’s %f
hypergeometric
function, the function hypergeometric
is a simplifying
function; also, hypergeometric
supports complex double and
big floating point evaluation. For the Gauss hypergeometric function,
that is \(p = 2\) and \(q = 1\), floating point evaluation
outside the unit circle is supported, but in general, it is not
supported.
When the option variable expand_hypergeometric
is true (default
is false) and one of the arguments a1
through ap
is a
negative integer (a polynomial case), hypergeometric
returns an
expanded polynomial.
Examples:
(%i1) hypergeometric([],[],x); (%o1) %e^x
Polynomial cases automatically expand when expand_hypergeometric
is true:
(%i2) hypergeometric([-3],[7],x); (%o2) hypergeometric([-3],[7],x) (%i3) hypergeometric([-3],[7],x), expand_hypergeometric : true; (%o3) -x^3/504+3*x^2/56-3*x/7+1
Both double float and big float evaluation is supported:
(%i4) hypergeometric([5.1],[7.1 + %i],0.42); (%o4) 1.346250786375334 - 0.0559061414208204 %i (%i5) hypergeometric([5,6],[8], 5.7 - %i); (%o5) .007375824009774946 - .001049813688578674 %i (%i6) hypergeometric([5,6],[8], 5.7b0 - %i), fpprec : 30; (%o6) 7.37582400977494674506442010824b-3 - 1.04981368857867315858055393376b-3 %i
Next: Parabolic Cylinder Functions, Previous: Struve Functions, Up: Special Functions [Contents][Index]