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LerchPhi

general Lerch Phi function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

LerchPhi(z, a, v)

Parameters

z

-

algebraic expression

a

-

algebraic expression

v

-

algebraic expression

Description

• 

The Lerch Phi function is defined as follows:

LerchPhiz,a,v=n=0znv+na

  

This definition is valid for z<1 or z=1and1<a. By analytic continuation, it is extended to the whole complex z-plane for each value of a and v.

• 

If v and a are positive integers, LerchPhi(z, a, v) has a branch cut in the z-plane along the real axis to the right of z&equals;1, with a branch point at z&equals;1.

• 

If a is a non-positive integer, LerchPhi(z, a, v) is a rational function of z with a pole of order 1a at z&equals;1.

• 

LerchPhi(1,a,v) = Zeta(0,a,v).  If 1<a, it is also true that limit(LerchPhi(z,a,v),z=1) = Zeta(0,a,v). If a1, this limit does not exist.

• 

If 0a, LerchPhi(z, a, v) has an infinite singularity at each non-positive integer v.

• 

If the coefficients of the series representation of a hypergeometric function are rational functions of the summation indices, then the hypergeometric function can be expressed as a linear sum of Lerch Phi functions.

• 

If the parameters of the hypergeometric functions are rational, we can express the hypergeometric function as a linear sum of polylog functions.

Examples

LerchPhi3&comma;4&comma;1

13polylog4&comma;3

(1)

LerchPhi0&comma;7&comma;4

116384

(2)

LerchPhi4&comma;0&comma;3

13

(3)

LerchPhiz&comma;a&comma;1

polyloga&comma;zz

(4)

LerchPhi1&comma;z&comma;1

&zeta;z

(5)

&DifferentialD;&DifferentialD;zLerchPhiz&comma;3&comma;4

LerchPhiz&comma;2&comma;4z4LerchPhiz&comma;3&comma;4z

(6)

References

  

Erdelyi, A. Higher Transcendental Functions. McGraw-Hill, 1953.

See Also

hypergeom

polylog

Zeta

 


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