meijerG

Meijer G-function

Syntax

meijerG(a,b,c,d,z)

Description

meijerG(a,b,c,d,z) returns the Meijer G-Function. meijerG is element-wise in z. The input parameters a, b, c, and d are vectors that can be empty, as in meijerG([], [], 3.2, [], 1).

Examples

Calculate Meijer G-Function for Numeric Inputs

syms x
meijerG(3, [], [], 2, 5)

ans =
    25

Call meijerG when z is an array. meijerG acts element-wise.

a = 2;
z = [1 2 3];
meijerG(a, [], [], [], z)

ans =
    0.3679    1.2131    2.1496

Calculate Meijer G-Function for Symbolic Numbers

Convert numeric input to symbolic form using sym, and find the Meijer G-function. For certain symbolic inputs, meijerG returns exact symbolic output using other functions.

meijerG(sym(2), [], [], [], sym(3))

ans =
3*exp(-1/3)

meijerG(sym(2/5), [], sym(1/2), [], sym(3))

ans =
(2^(4/5)*3^(1/2)*gamma(1/10))/80

Find Meijer G-Function for Symbolic Variables or Expressions

For symbolic variables or expressions, meijerG returns an output in terms of simple or special functions.

syms a b c d z
f = meijerG(a,b,c,d,z)

f =
(gamma(c - a + 1)*(1/z)^(1 - a)*hypergeom([c - a + 1, d - a + 1],...
 b - a + 1, 1/z))/(gamma(b - a + 1)*gamma(a - d))

Substitute values for the variables by using subs, and convert values to double by using double.

fVal = subs(f, [a b c d z], [1.2 3 5 7 9])

fVal =
(266*9^(1/5)*hypergeom([24/5, 34/5], 14/5, 1/9))/(25*gamma(-29/5))

double(fVal)

ans =
   5.7586e+03

Calculate fVal to higher precision using vpa.

vpa(fVal)

ans =
5758.5946416377834597597497022199

Relations Between Meijer G-Function and Other Functions

Show relations between meijerG and simpler functions for given parameter values.

Show that when a, b, and d are empty, and c = 0, then meijerG reduces to exp(-z).

syms z
meijerG([], [], 0, [], z)

ans =
exp(-z)

Show that when a, b, and d are empty, and c = [1/2 -1/2], then meijerG reduces to 2K_v(1,2z^1/2).

meijerG([], [], [1/2 -1/2], [], z)

ans =
2*besselk(1, 2*z^(1/2))

Plot Meijer G-Function

Plot the real and imaginary values of the Meijer G-function for values of b and z, where a = [-2 2] and c and d are empty. Fill the contours by setting Fill to on.

syms b z
f = meijerG([-2 2], b, [], [], z);

subplot(2,2,1)
fcontour(real(f),'Fill','on')
title('Real Values of Meijer G')
xlabel('b')
ylabel('z')

subplot(2,2,2)
fcontour(imag(f),'Fill','on')
title('Imaginary Values of Meijer G')
xlabel('b')
ylabel('z')

Input Arguments

a — Input

number | vector | symbolic number | symbolic variable | symbolic vector | symbolic function | symbolic expression

Input, specified as a number or vector, or a symbolic number, variable, vector, function, or expression.

b — Input

number | vector | symbolic number | symbolic variable | symbolic vector | symbolic function | symbolic expression

Input, specified as a number or vector, or a symbolic number, variable, vector, function, or expression.

c — Input

number | vector | symbolic number | symbolic variable | symbolic vector | symbolic function | symbolic expression

Input, specified as a number or vector, or a symbolic number, variable, vector, function, or expression.

d — Input

number | vector | symbolic number | symbolic variable | symbolic vector | symbolic function | symbolic expression

Input, specified as a number or vector, or a symbolic number, variable, vector, function, or expression.

z — Input

number | vector | symbolic number | symbolic variable | symbolic vector | symbolic matrix | symbolic multidimensional array | symbolic function | symbolic expression

Input, specified as a number or vector, or a symbolic number, variable, vector, function, or expression.

More About

Meijer G-Function

The Meijer G-function meijerG([a1,…,an], [an + 1,…,ap], [b1,…,bm], [bm + 1,…,bq], z) is a general function that includes other special functions as particular cases, and is defined as

$$G_{p,q}^{m,n}\left(\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}|z\right)=\frac{1}{2\pi \text{i}}{\displaystyle \int \frac{\left({\displaystyle \prod }_{j=1}^m\Gamma \left(b_j-s\right)\right)\left({\displaystyle \prod }_{j=1}^n\Gamma \left(1-a_j+s\right)\right)}{\left({\displaystyle \prod }_{j=m+1}^q\Gamma \left(1-b_j+s\right)\right)\left({\displaystyle \prod }_{j=n+1}^p\Gamma \left(a_j-s\right)\right)}}{z}^{s}ds.$$

Algorithms

For the Meijer G-function meijerG([a1,…,an], [an + 1,…,ap], [b1,…,bm], [bm + 1,…,bq], z), for a_i ∊ (a₁,…,a_n) and b_j ∊ (b₁,…,b_m), no pair of parameters a_i − b_j should differ by a positive integer.

The Meijer G-function involves a complex contour integral with one of the following types of integration paths:

• The contour goes from - i ∞ to i ∞ so that all poles of $$\Gamma \left(b_j-s\right)$$, j = 1, …, m lie to the right of the path, and all poles of $$\Gamma \left(1-a_k+s\right)$$, k = 1, …, n lie to the left of the path. The integral converges if $$c=m+n-\frac{p+q}{2}>0$$, |arg(z)| < c π. If |arg(z)| = c π, c ≥ 0, the integral converges absolutely when p = q and ℜ(ψ) < - 1, where $$\Psi =\left({\displaystyle \sum }_{j=1}^q{b}_j\right)-\left({\displaystyle \sum }_{i=1}^p{a}_i\right)$$. When p ≠ q, the integral converges if you choose the contour so that the contour points near i ∞ and - i ∞ have a real part σ satisfying $$\left(q-p\right)\sigma >\Re (\psi )+1-\frac{q-p}{2}$$.

• The contour is a loop beginning and ending at ∞ and encircling all poles of $$\Gamma \left(b_j-s\right)$$, j = 1, …, m moving in the negative direction, but none of the poles of $$\Gamma \left(1-a_k+s\right)$$, k = 1, …, n. The integral converges if q ≥ 1 and either p < q or p = q and |z| < 1.

• The contour is a loop beginning and ending at - ∞ and encircling all poles of $$\Gamma \left(1-a_k+s\right)$$, k = 1, …, n moving in the positive direction, but none of the poles of $$\Gamma \left(b_j+s\right)$$, j = 1, …, m. The integral converges if p ≥ 1 and either p > q or p = q and |z| > 1.

The integral represents an inverse Laplace transform or, more specifically, a Mellin-Barnes type of integral.

For a given set of parameters, the contour chosen in the definition of the Meijer G-function is the one for which the integral converges. If the integral converges for several contours, all contours lead to the same function.

The Meijer G-function satisfies a differential equation of order max(p, q) with respect to a variable z:

$$\left({\left(-1)\right)}^{m+n-p}z\left({\displaystyle \prod }_{i=1}^p\left(z\frac{d}{dz}-a_i-1\right)\right)-{\displaystyle \prod _{j=1}^q\left(z\frac{d}{dz}-b_j\right)}\right)G_{p,q}^{m,n}\left(\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_p\end{array}|z\right)=0.$$

If p < q, this differential equation has a regular singularity at z = 0 and an irregular singularity at z = ∞. If p = q, the points z = 0 and z = ∞ are regular singularities, and there is an additional regular singularity at z = (−1)^{m + n - p}.

The Meijer G-function represents an analytic continuation of the hypergeometric function [1]. For particular choices of parameters, you can express the Meijer G-function through the hypergeometric function. For example, if no two of the b_h terms, h = 1, …, m, differ by an integer or zero and all poles are simple, then

$$G_{p,q}^{m,n}\left(\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_p\end{array}|z\right)={\displaystyle \sum _{h=1}^m\frac{\left({\displaystyle \prod _{j=1\dots m,j\ne h}\Gamma }\left(b_j-b_h\right)\right)\left({\displaystyle \prod _{j=1}^n\Gamma }\left(1+b_h-a_j\right)\right)}{\left({\displaystyle \prod _{j=m+1}^q\Gamma }\left(1+b_h-b_j\right)\right)\left({\displaystyle \prod _{j=n+1}^p\Gamma }\left(a_j-b_h\right)\right)}}{z}^b{h}_p{F}_{q-1}\left(A_h;B_h;{\left(-1\right)}^{p-m-n}z\right).$$

Here p < q or p = q and |z| < 1. A_h denotes

$$A_h=1+b_h-a_1,\dots ,1+b_h-a_p.$$

B_h denotes

$$B_h=1+b_h-b_1,\dots ,1+b_h-b_{(h-1)},1+b_h-b_{h+1},\dots ,1+b_h-b_q.$$

Meijer G-functions with different parameters can represent the same function.

• The Meijer G-function is symmetric with respect to the parameters. Changing the order inside each of the following lists of vectors does not change the resulting Meijer G-function: [a₁, …, a_n], [a_{n + 1}, …, a_p], [b₁, …, b_m], [b_{m + 1}, …, b_q].

• If z is not a negative real number and z ≠ 0, the function satisfies the following identity:

  $$G_{p,q}^{m,n}\left(\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}|z\right)=G_{q,p}^{n,m}\left(\begin{array}{c}1-b_1,\dots ,1-b_p\\ 1-a_1,\dots ,1-a_p\end{array}|\frac{1}{z}\right).$$

  .

• If 0 < n < p and r = a₁ - a_p is an integer, the function satisfies the following identity:

  $$G_{p,q}^{m,n}\left(\begin{array}{c}{a}_1,a_2,\dots ,a_{p-1},a_p\\ b_1,b_2,\dots ,b_{q-1},b_q\end{array}|z\right)=G_{p,q}^{m,n}\left(\begin{array}{c}{a}_p,a_2,\dots ,a_{p-1},a_1\\ b_1,b_2,\dots ,b_{q-1},b_q\end{array}|z\right).$$

  .

• If 0 < m < q and r = b₁ - b_q is an integer, the function satisfies the following identity:

  $$G_{p,q}^{m,n}\left(\begin{array}{c}{a}_1,a_2,\dots ,a_{p-1},a_p\\ b_1,b_2,\dots ,b_{q-1},b_q\end{array}|z\right)={\left(-1\right)}^{\gamma }{G}_{p,q}^{m,n}\left(\begin{array}{c}{a}_1,a_2,\dots ,a_{p-1},a_p\\ b_q,b_2,\dots ,b_{q-1},b_1\end{array}|z\right).$$

  .

According to these rules, the meijerG function call can return meijerG with modified input parameters.

References

[1] Luke, Y. L., The Special Functions and Their Approximations. Vol. 1. New York: Academic Press, 1969.

[2] Prudnikov, A. P., Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol 3: More Special Functions. Gordon and Breach, 1990.

[3] Abramowitz, M., I. A. Stegun, Handbook of Mathematical Functions. 9th printing. New York: Dover Publications, 1970.

Version History

Introduced in R2017b