Kelvin functions ber and bei - Maple Help

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KelvinBer, KelvinBei - Kelvin functions ber and bei

KelvinKer, KelvinKei - Kelvin functions ker and kei

KelvinHer, KelvinHei - Kelvin functions her and hei

Calling Sequence

KelvinBer(v, x)

KelvinBei(v, x)

KelvinKer(v, x)

KelvinKei(v, x)

KelvinHer(v, x)

KelvinHei(v, x)

Parameters

v

-

algebraic expression (the order or index)

x

-

algebraic expression (the argument)

Description

• 

The Kelvin functions (sometimes known as the Thomson functions) are defined by the following equations:

KelvinBerv,x+IKelvinBeiv,x=BesselJv,x122+12I2

KelvinBerv,xIKelvinBeiv,x=BesselJv,x12212I2

KelvinKerv,x+IKelvinKeiv,x=ⅇ12IvπBesselKv,x122+12I2

KelvinKerv,xIKelvinKeiv,x=ⅇ12IvπBesselKv,x12212I2

KelvinHerv,x+IKelvinHeiv,x=HankelH1v,x122+12I2

KelvinHerv,xIKelvinHeiv,x=HankelH2v,x12212I2

• 

The Kelvin functions are all real valued for real x and positive v.

Examples

KelvinBer0,0

1

(1)

KelvinKei1.5I,2.6+3I

0.081603765080.03651099032I

(2)

seriesKelvinHer1,x,x,3

2πx+142ln1414Ix2ln14+14Ix22γ+Iln1414Ix2Iln14+14Ix2+1ππx+Ox3

(3)

convertKelvinBeiv,x,BesselJ

12IBesselJv,1212Ix2BesselJv,12+12Ix2

(4)

xKelvinHeiv,x

122KelvinHeiv+1,xKelvinHerv+1,x+vKelvinHeiv,xx

(5)

convertKelvinBerv,x,BesselJ

12BesselJv,1212Ix2+12BesselJv,12+12Ix2

(6)

convertKelvinBeiv,x,Bessel

12IBesselJv,1212Ix2BesselJv,12+12Ix2

(7)

convertKelvinKerv,x,BesselK

12BesselKv,12+12Ix2+ⅇ12Ivπ2BesselKv,1212Ix2ⅇ12Ivπ

(8)

convertKelvinHerv,x,Hankel

12HankelH1v,12+12Ix2+12HankelH2v,1212Ix2

(9)

See Also

Airy, Anger, Bessel, convert/Bessel, inifcns, Struve

References

  

Abramowitz, M., and Stegun, I. Handbook of Mathematical Functions, Section 9.9. Washington: National Bureau of Standards Applied Mathematics, 1964.

  

Erdelyi, A., ed. Higher Transcendental Functions, Section 7.2.3. New York: McGraw-Hill, 1953.


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