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Psi

the Digamma and Polygamma functions

 

Calling Sequence

Parameters

Description

Examples

References

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)

References

  

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

See Also

expand

GAMMA

initialfunctions

Zeta

 


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