the Digamma and Polygamma functions - Maple Help

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Psi - the Digamma and Polygamma functions

Calling Sequence

Psi(x)

Ψx

Psi(n,x)

Ψn,x

Parameters

x

-

expression

n

-

expression

Description

• 

Psi(x) is the digamma function,

Ψx=ⅆⅆxlnΓx=ⅆⅆxΓxΓx

• 

Psi(n, x) is the nth polygamma function, which is the nth derivative of the digamma function.

• 

You can enter the command Psi using either the 1-D or 2-D calling sequence.

• 

If n is an integer greater than one, Psi(n) + gamma is a rational number. (gamma is Euler's constant.) For small values of n, Psi(n) computes as a sum of gamma and a rational number. To perform this computation for larger values of n, use expand.

Ψn,x=ⅆnⅆxnΨx

Ψ0,x=Ψx

• 

Psi(n, x) is extended to complex n, including negative integer indices, by the formula

Ψw,z=ζ1,w+1,z+γ+Ψwζ0,w+1,zΓw

  

where ζ is the Hurwitz zeta function.

Examples

Ψ2

1γ

(1)

Ψ1,2

1+16π2

(2)

Ψ3.5+4.7I

1.717883835+1.001470255I

(3)

Ψ7,2.2+3.3I

0.02713341434+0.003825068416I

(4)

Ψ2,1.543

0.7957394716

(5)

Ψ1.342+I,3.5233

0.69889190050.7978763419I

(6)

Ψ50

138812566871391350266313099044504245996706400γ

(7)

Ψ51

Ψ51

(8)

Evaluating Psi(51) directly is faster than expanding and then evaluating.

expandΨ51

γ+139432375772240549607593099044504245996706400

(9)

evalf

3.921989673

(10)

evalfΨ51

3.921989673

(11)

See Also

expand, GAMMA, initialfunctions, Zeta

References

  

Espinosa, O., and Moll, V. "A Generalized Polygamma Function." Integral Transforms and Special Functions, (April 2004): 101-115.


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