# MarcumQ

MarcumQ[m,a,b]

gives Marcum's Q function .

MarcumQ[m,a,b0,b1]

gives Marcum's Q function .

# Details

• Mathematical function, suitable for both symbolic and numerical evaluation.
• 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(27)

### Numerical Evaluation(4)

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:

### 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:

### 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)

Wolfram Research (2010), MarcumQ, Wolfram Language function, https://reference.wolfram.com/language/ref/MarcumQ.html.

#### Text

Wolfram Research (2010), MarcumQ, Wolfram Language function, https://reference.wolfram.com/language/ref/MarcumQ.html.

#### CMS

Wolfram Language. 2010. "MarcumQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MarcumQ.html.

#### APA

Wolfram Language. (2010). MarcumQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MarcumQ.html

#### BibTeX

@misc{reference.wolfram_2023_marcumq, author="Wolfram Research", title="{MarcumQ}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MarcumQ.html}", note=[Accessed: 26-February-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_marcumq, organization={Wolfram Research}, title={MarcumQ}, year={2010}, url={https://reference.wolfram.com/language/ref/MarcumQ.html}, note=[Accessed: 26-February-2024 ]}