MarcumQ

MarcumQ[m,a,b]

gives Marcum's Q function TemplateBox[{m, a, b}, MarcumQ].

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  (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 :

TemplateBox[{m, a, x}, MarcumQ] is an even function of :

TemplateBox[{m, a, b}, MarcumQ] 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)

TemplateBox[{m, a, b}, MarcumQ] can be expressed in terms of simpler functions whenever is a half-integer:

For integer and , TemplateBox[{m, a, b}, MarcumQ] can be expressed in terms of modified Bessel functions:

For arbitrary and , TemplateBox[{m, a, b}, MarcumQ] can be expressed in terms of hypergeometric functions:

Ordinary differential equation with respect to satisfied by TemplateBox[{m, a, b}, MarcumQ]:

Ordinary differential equation with respect to satisfied by TemplateBox[{m, a, b}, MarcumQ]:

Recurrence relation with respect to satisfied by TemplateBox[{m, a, b}, MarcumQ]:

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_2024_marcumq, author="Wolfram Research", title="{MarcumQ}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MarcumQ.html}", note=[Accessed: 19-June-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_marcumq, organization={Wolfram Research}, title={MarcumQ}, year={2010}, url={https://reference.wolfram.com/language/ref/MarcumQ.html}, note=[Accessed: 19-June-2024 ]}