Calling psi for a number that is not a symbolic object invokes the MATLAB psi function. This function accepts real arguments only. 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).

Input Arguments

`x`	Nonnegative symbolic number, variable, or expression.
`k`	Nonnegative integer.
`A`	Vector or matrix of nonnegative symbolic numbers, variables, or expressions.

Definitions

digamma Function

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

polygamma Function

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

Examples

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))]

The results are:

ans =
   -1.9635  -16.8288   -0.1248    2.0372

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 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 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)

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 the digamma function for elements of these 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