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convert/Kelvin

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

 

Calling Sequence

Parameters

Description

Examples

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

3zKelvinKer13&comma;13I3z322+IKelvinKei13&comma;13I3z3223+I6π

(3)

Consider the following expression

solyz&equals;_C1KummerM12&plus;a&comma;1&plus;2a&comma;z&plus;_C2KummerU12&plus;a&comma;1&plus;2a&comma;z

solyz=_C1KummerM12+a&comma;1+2a&comma;z+_C2KummerU12+a&comma;1+2a&comma;z

(4)

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

PDEtoolsdpolyformsol&comma;no_Fn

&DifferentialD;2&DifferentialD;z2yz=2a+z1&DifferentialD;&DifferentialD;zyzz+1+2ayz2z&where

(5)

sol can be rewritten using Kelvin functions

convertsol&comma;Kelvin

yz=_C1&ExponentialE;z2Γ1+aKelvinBera&comma;14I4z2+IKelvinBeia&comma;14I4z22aI2za+_C22aKelvinBera&comma;14I4z24I2za21+2aI2za21+2aKelvinBera&comma;14I4z24z2a2aIKelvinBeia&comma;14I4z2I2za21+2a4z2a2a+IKelvinBeia&comma;14I4z22a4I2za21+2aπ&ExponentialE;z2sinπ1+a

(6)

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

yz=&ExponentialE;z22_C1Γ1+asinπa4a_C2πIzaKelvinBera&comma;14I4z22sinπa_C2z2aIzaπ&ExponentialE;z2KelvinBera&comma;14I4z22sinπ1+a+I&ExponentialE;z2Iza22+2a_C1Γ1+asinπa2_C2πKelvinBeia&comma;14I4z24sinπaI_C2z2aIzaπ&ExponentialE;z2KelvinBeia&comma;14I4z22sinπ1+a

(7)

See Also

convert

convert/to_special_function

FunctionAdvisor