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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

Parameters

Description

Examples

References

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)

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.

See Also

Airy

Anger

Bessel

convert/Bessel

inifcns

Struve

 


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