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.
  • Q_m(a,b)=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 a certain special argument, 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  (16)

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  (2)

MarcumQ for symbolic a:

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

Visualization  (2)

Plot the MarcumQ function for various parameters:

Plot the real part of :

Plot the imaginary part of :

Function Properties  (4)

Real domain of MarcumQ:

Complex domain of MarcumQ:

Approximate function range of MarcumQ:

MarcumQ is an even function:

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:

Applications  (2)

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)

Introduced in 2010
 (8.0)