convert special functions admitting 1F1 or 0F1 hypergeometric representation into Kelvin functions - Maple Help

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convert/Kelvin - convert special functions admitting 1F1 or 0F1 hypergeometric representation into Kelvin functions

Calling Sequence

convert(expr, Kelvin)

Parameters

expr

-

Maple expression, equation, or a set or list of them

Description

• 

convert/Kelvin converts, when possible, special functions admitting a 1F1 or 0F1 hypergeometric representation into Kelvin functions. The Kelvin functions are

FunctionAdvisor( Kelvin );

The 6 functions in the "Kelvin" class are:

KelvinBei,KelvinBer,KelvinHei,KelvinHer,KelvinKei,KelvinKer

(1)

Examples

AiryAiz

AiryAiz

(2)

convert&comma;Kelvinassuming0<&real;z

163zKelvinKer13&comma;1313Iz3&sol;22&plus;IKelvinKei13&comma;1313Iz3&sol;223&plus;I&pi;

(3)

Consider the following expression

sol:=yz&equals;_C1KummerM12&plus;a&comma;1&plus;2a&comma;z&plus;_C2KummerU12&plus;a&comma;1&plus;2a&comma;z

sol:=yz&equals;_C1KummerM12&plus;a&comma;1&plus;2a&comma;z&plus;_C2KummerU12&plus;a&comma;1&plus;2a&comma;z

(4)

where sol is the solution to the following linear differential equation (see dpolyform)

PDEtools&lsqb;dpolyform&rsqb;sol&comma;no_Fn

&DifferentialD;2&DifferentialD;z2yz&equals;2a&plus;z1&DifferentialD;&DifferentialD;zyzz&plus;121&plus;2ayzz &where

(5)

sol can be rewritten using Kelvin functions

convertsol&comma;Kelvin

yz&equals;_C1&ExponentialE;12z&Gamma;1&plus;aKelvinBera&comma;1414Iz2&plus;IKelvinBeia&comma;1414Iz22a12Iza&plus;_C2142aKelvinBera&comma;1414Iz212Iza21&plus;2a1412Iza21&plus;2aKelvinBera&comma;1414Iz2z2a2a14IKelvinBeia&comma;1414Iz212Iza21&plus;2az2a2a&plus;14IKelvinBeia&comma;1414Iz22a12Iza21&plus;2a&pi;&ExponentialE;12zsin&pi;1&plus;a

(6)

collect&comma;KelvinBer&comma;KelvinBei&comma;simplify

yz&equals;12&ExponentialE;12z2_C1&Gamma;aasin&pi;a4a_C2&pi;IzaKelvinBera&comma;1414Iz2sin&pi;a12_C2z2aIza&pi;&ExponentialE;12zKelvinBera&comma;1414Iz2sin&pi;1&plus;a&plus;14I&ExponentialE;12zIza22&plus;2a_C1&Gamma;aasin&pi;a2_C2&pi;KelvinBeia&comma;1414Iz2sin&pi;a12I_C2z2aIza&pi;&ExponentialE;12zKelvinBeia&comma;1414Iz2sin&pi;1&plus;a

(7)

See Also

convert, convert/to_special_function, FunctionAdvisor


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