GATE-CS-2014 (Set-2) Question 1

This is a coding problem in which you have to implement a function gamma, which takes as input two integers a and b and returns the value of the mathematical function gamma(a, b), defined as:

γ(a, b) = ∫[0,∞] t^(a-1) * e^(-t) * t^(b-1) dt

where ∫[0,∞] is the definite integral from 0 to infinity.

Approach

We can use the gamma function to compute γ(a, b). The gamma function is defined as:

Γ(a) = ∫[0,∞] t^(a-1) * e^(-t) dt

We can write the integral γ(a, b) in terms of the gamma function as:

γ(a, b) = Γ(a) * Γ(b) / Γ(a + b)

We can use numerical integration to compute the gamma function Γ(a) and then use the above formula to compute γ(a, b).

We can use the Simpson's rule for numerical integration. Simpson's rule is an improvement over the trapezoidal rule and provides a more accurate estimation of the integral.

Code

from math import gamma
def simpson(f, a, b, n=1000):
    h = (b-a)/n
    s = f(a) + f(b)
    for i in range(1, n, 2):
        s += 4 * f(a + i * h)
    for i in range(2, n-1, 2):
        s += 2 * f(a + i * h)
    return s * h / 3

def gamma(a, b):
    return simpson(lambda t: t**(a-1) * math.exp(-t) * t**(b-1), 0, math.inf) * math.gamma(a) * math.gamma(b) / math.gamma(a+b)

In the above code, we define a function simpson to implement Simpson's rule for numerical integration. We then define the function gamma that uses Simpson's rule to compute the gamma function and then uses the formula for γ(a, b) to compute the required value. We use the math module to access the gamma function for computing the gamma function.