psi

Digamma function

Syntax

psi(x)

psi(k,x)

Description

psi(x) computes the digamma function of x.

psi(k,x) computes the polygamma function of x, which is the kth derivative of the digamma function at x.

Examples

Compute Digamma and Polygamma for Numeric Inputs

Compute the digamma and polygamma functions for these numbers. Because these numbers are not symbolic objects, you get the floating-point results.

[psi(1/2) psi(2, 1/2) psi(1.34) psi(1, sin(pi/3))]

ans =
   -1.9635  -16.8288   -0.1248    2.0372

Compute Digamma and Polygamma for Symbolic Inputs

Compute the digamma and polygamma functions for the numbers converted to symbolic objects.

[psi(sym(1/2)), psi(1, sym(1/2)), psi(sym(1/4))]

ans =
[ - eulergamma - 2*log(2), pi^2/2, - eulergamma - pi/2 - 3*log(2)]

For some symbolic (exact) numbers, psi returns unresolved symbolic calls.

psi(sym(sqrt(2)))

ans =
psi(2^(1/2))

Compute Derivatives of Digamma and Polygamma Functions

Compute the derivatives of these expressions containing the digamma and polygamma functions.

syms x
diff(psi(1, x^3 + 1), x)
diff(psi(sin(x)), x)

ans =
3*x^2*psi(2, x^3 + 1)
 
ans =
cos(x)*psi(1, sin(x))

Expand Digamma and Polygamma Functions

Expand the expressions containing the digamma functions.

syms x
expand(psi(2*x + 3))
expand(psi(x + 2)*psi(x))

ans =
psi(x + 1/2)/2 + log(2) + psi(x)/2 +...
1/(2*x + 1) + 1/(2*x + 2) + 1/(2*x)
 
ans =
psi(x)/x + psi(x)^2 + psi(x)/(x + 1)

Limit of Digamma and Polygamma Functions

Compute the limits for expressions containing the digamma and polygamma functions.

syms x
limit(x*psi(x), x, 0)
limit(psi(3, x), x, inf)

ans =
-1
 
ans =
0

Compute Digamma for Matrix Input

Compute the digamma function for elements of matrix M and vector V.

M = sym([0 inf; 1/3 1/2]);
V = sym([1, inf]);
psi(M)
psi(V)

ans =
[                                          Inf,                     Inf]
[ - eulergamma - (3*log(3))/2 - (pi*3^(1/2))/6, - eulergamma - 2*log(2)]

ans =
[ -eulergamma, Inf]

Compute Polygamma for Matrix Input

Compute the polygamma function for elements of matrix M and vector V. The psi function acts elementwise on nonscalar inputs.

M = sym([0 inf; 1/3 1/2]);
polyGammaM = [1 3; 2 2];
V = sym([1, inf]);
polyGammaV = [6 6];
psi(polyGammaM,M)
psi(polyGammaV,V)

ans =
[                               Inf,           0]
[ - 26*zeta(3) - (4*3^(1/2)*pi^3)/9, -14*zeta(3)]
 
ans =
[ -720*zeta(7), 0]

Because all elements of polyGammaV have the same value, you can replace polyGammaV by a scalar of that value. psi expands the scalar into a nonscalar of the same size as V and computes the result.

V = sym([1, inf]);
psi(6,V)

ans =
[ -720*zeta(7), 0]

Input Arguments

x — Input

symbolic number | symbolic variable | symbolic expression | symbolic array

Input, specified as a symbolic number, variable, expression, or array, or expression.

k — Input

nonnegative integer | nonnegative integer or vector, matrix or multidimensional array of nonnegative integers.

Input, specified as a nonnegative integer or vector, matrix or multidimensional array of nonnegative integers. If x is nonscalar and k is scalar, then k is expanded into a nonscalar of the same dimensions as x with each element being equal to k. If both x and k are nonscalars, they must have the same dimensions.

More About

Digamma Function

The digamma function is the first derivative of the logarithm of the gamma function:

$$\psi \left(x\right)=\frac{d}{dx}\ln \Gamma \left(x\right)=\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}$$

Polygamma Function

The polygamma function of the order k is the (k + 1)th derivative of the logarithm of the gamma function:

$${\psi }^{\left(k\right)}\left(x\right)=\frac{d^{k+1}}{d{x}^{k+1}}\ln \Gamma \left(x\right)=\frac{d^k}{d{x}^k}\psi \left(x\right)$$

Tips

• Calling psi for a number that is not a symbolic object invokes the MATLAB^® psi function. This function accepts real nonnegative arguments x. If you want to compute the polygamma function for a complex number, use sym to convert that number to a symbolic object, and then call psi for that symbolic object.

• psi(0, x) is equivalent to psi(x).