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MarcumQ

MarcumQ[m,a,b]

gives Marcum's Q function .

MarcumQ[m,a,b₀,b₁]

gives Marcum's Q function $TemplateBox[{m, a, {b, _, 0}}, MarcumQ]-TemplateBox[{m, a, {b, _, 1}}, MarcumQ]$ .

Details

Mathematical function, suitable for both symbolic and numerical evaluation.
$TemplateBox[{m, a, b}, MarcumQ]=int_b^inftyx (x/a)^(m-1) TemplateBox[{{m, -, 1}, {a, , x}}, BesselI] exp(-1/2 (a^2+x^2))dx$ for real positive , , and .
MarcumQ[m,a,b] is an entire function of both a and b with no branch cut discontinuities.
For certain special arguments, MarcumQ automatically evaluates to exact values.
MarcumQ can be evaluated to arbitrary numerical precision.
MarcumQ automatically threads over lists. »

Examples

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Basic Examples (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope (29)

Numerical Evaluation (6)

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:

Specific Values (3)

MarcumQ for symbolic a:

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

The four-argument form gives the difference:

Visualization (2)

Plot the MarcumQ function for various parameters:

Plot the real part of :

Plot the imaginary part of :

Function Properties (9)

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:

Differentiation (2)

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:

Series Expansions (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Function Identities and Simplifications (5)

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 :

Applications (2)

The amplitude of a signal is modeled by RiceDistribution. Find the probability that the amplitude will exceed its mean value:

Evaluate numerically:

Compare the value of the MarcumQ function for large arguments to its asymptotic formula:

Construct an approximation using the central limit theorem:

Evaluate numerically:

Properties & Relations (3)

MarcumQ can be used to compute the SurvivalFunction of SkellamDistribution:

MarcumQ computes the SurvivalFunction of NoncentralChiSquareDistribution:

MarcumQ computes the SurvivalFunction of RiceDistribution:

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