Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix MarcumQ function using MatrixFunction:

Compute average-case statistical intervals using Around:

MarcumQ for symbolic a:

Find the maximum of MarcumQ[1,2,x]:

The four-argument form gives the difference:

Plot the MarcumQ function for various parameters:

Plot the real part of :

Plot the imaginary part of :

Real domain of MarcumQ:

Complex domain of MarcumQ:

Approximate function range of :

is an even function of :

is an analytic function of and for positive integer :

It has no singularities or discontinuities:

is neither non-increasing nor non-decreasing:

is not injective:

is non-negative:

is neither convex nor concave:

TraditionalForm formatting:

First derivative with respect to a:

First derivative with respect to b:

Higher derivatives with respect to a:

Plot the higher derivatives with respect to a when b=3 and m=1:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

can be expressed in terms of simpler functions whenever is a half-integer:

For integer and , can be expressed in terms of modified Bessel functions:

For arbitrary and , can be expressed in terms of hypergeometric functions:

Ordinary differential equation with respect to satisfied by :

Ordinary differential equation with respect to satisfied by :

Recurrence relation with respect to satisfied by :