Python program to calculate logarithmic gamma of a given number

In mathematics, the gamma function is considered to be the expansion of the factorial of any given number. However, since factorial is defined only for real numbers, the gamma function is beyond the scope of defining factorial for all complex numbers except negative integers. It is represented by -

Γ(x) = (x-1)!

Logarithmic Gamma Function appears because the gamma function only grows rapidly at larger numbers, so applying the logarithm to gamma will slow it down a lot. It is also known as the natural logarithm gamma of a given number.

log(Γ(x)) = log((x-1)!)

In the Python programming language, like other programming languages, the logarithmic gamma function is calculated using the math.lgamma() function. However, we will also look at a few other ways to calculate the log gamma of a number in this article.

Input and output scenarios

Let's look at some input and output scenarios to find the log-gamma function using the math.lgamma() method.

Assume that the input of the logarithmic gamma function is a positive integer -

Input: 12
Result: 17.502307845873887

Assume that the input of the logarithmic gamma function is a negative integer -

Input: -12
Result: “ValueError: math domain error”

Assume that the input to the log gamma function is zero -

Input: 0
Result: “ValueError: math domain error”

Assume that the input to the log gamma function is a negative decimal value close to zero -

Input: -0.2
Result: 1.761497590833938

A domain error occurs when using the lgamma() method because the function is defined for all complex numbers minus negative "integers". Let's look at various ways to find the log gamma of a given number.

Usemath.lgamma()Function

The lgamma() method is defined in the math library and returns the natural logarithm gamma value of a given number. The syntax of this method is -

math.lgamma(x)

where x is any complex number except negative integers.

Example

The Python example of using the math.lgamma() function to find log gamma is as follows -

# import math library
import math

#log gamma of positive integer
x1 = 10
print(math.lgamma(x1))

#log gamma of negative complex number
x2 = -1.2
print(math.lgamma(x2))

#log gamma of a positive complex number
x3 = 3.4
print(math.lgamma(x3))

Output

The output of the above python code is -

12.801827480081467
1.5791760340399836
1.0923280598027416

Using math.gamma() and math.log() functions

In another approach, the logarithm of a number can be found by first finding the gamma of the number using the math.gamma() function, and then applying the logarithm to the gamma value using gamma. b>math.log() function. Here, we just break the lgamma() function into steps.

Example

The python implementation of the above process is as follows -

# import math library
import math

#log gamma of positive integer
x1 = math.gamma(10)
print(math.log(x1))

#log gamma of negative complex number
x2 = math.gamma(-1.2)
print(math.log(x2))

#log gamma of a positive complex number
x3 = math.gamma(3.4)
print(math.log(x3))

Output

The output obtained is as follows -

12.801827480081469
1.5791760340399839
1.0923280598027414

By applying logarithms to the factorial of a number

An easier way is to find the factorial of a given number, since the gamma function is defined as the factorial of a complex number, and compute the factorial by applying the logarithm to it using the math.log() method.

Example

In this Python example, we use factorial and the math.log() method to find the logarithmic gamma of a number. The only disadvantage of using this method is that it only works with positive integers.

# import math library
import math

def factorial(n):
   if n == 1:
      return 1
   else:
      return n*factorial(n-1)
	
#log gamma of positive integer
x1 = 10
y1 = factorial(x1-1)
print(math.log(y1))

x2 = 3
y2 = factorial(x2-1)
print(math.log(y2))

#log gamma of a positive complex number
x3 = 3.4
y3 = factorial(x3-1)
print(math.log(y3))

Output

The output is -

12.801827480081469
0.6931471805599453
RecursionError: maximum recursion depth exceeded in comparison

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