MarcumQ

MarcumQ (m, a, b)

The MarcumQ function returns the generalized Q-function defined by the integral

$$\displaystyle Q_{m}(a, b)=\int_{b}^{\infty} u\left(\frac{u}{a}\right)^{m-1} \exp \left(-\frac{\left(a^{2}+u^{2}\right)}{2}\right) I_{m-1}(a u) d u,$$

where I_k is the modified Bessel function of the first kind and order k.

Its applications have been primarily in the fields of communication and detection theory. However, an interesting interpretation of its result with m=1 and appropriate parameter scaling is the fractional power of a two-dimensional circular Gaussian function within a displaced circular aperture.

Depending on the input arguments, the MarcumQ function may be computationally intensive but you can abort the calculation at any time.

References

Cantrell, P.E., and A.K. Ojha, Comparison of Generalized Q-Function Algorithms, IEEE Transactions on Information Theory, IT-33, 591-596, 1987.

Simon, M. K., A New Twist on the Marcum Q-Function and Its Application, IEEE Communications Letters, 3, 39-41, 1998.