MATLAB Function Reference

psi

Psi (polygamma) function

Syntax

• Y = psi(X)
  Y = psi(k,X)
  Y = psi(k0:k1,X)
  

Description

Y = psi(X) evaluates the function for each element of array X. X must be real and nonnegative. The function, also known as the digamma function, is the logarithmic derivative of the gamma function

Y = psi(k,X) evaluates the kth derivative of at the elements of X. psi(0,X) is the digamma function, psi(1,X) is the trigamma function, psi(2,X) is the tetragamma function, etc.

Y = psi(k0:k1,X) evaluates derivatives of order k0 through k1 at X. Y(k,j) is the (k-1+k0)th derivative of , evaluated at X(j).

Examples

Example 1. Use the psi function to calculate Euler's constant, .

• format long
  -psi(1)
  ans =
     0.57721566490153
  
  -psi(0,1)
  ans =
     0.57721566490153
  

Example 2. The trigamma function of 2, psi(1,2), is the same as .

• format long
  psi(1,2)
  ans =
     0.64493406684823
  
  pi^2/6 - 1
  ans =
     0.64493406684823
  

Example 3. This code produces the first page of Table 6.1 in Abramowitz and Stegun [1].

• x = (1:.005:1.250)';  
  [x gamma(x) gammaln(x) psi(0:1,x)' x-1]
  

Example 4. This code produces a portion of Table 6.2 in [1].

• psi(2:3,1:.01:2)'
  

See Also

gamma, gammainc, gammaln

References

[1]  Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publlications, 1965, Sections 6.3 and 6.4.

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