Factorial and gamma functions in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of factorial for a positive integer

What is the output of this code snippet using scipy's factorial function?

SciPy

from scipy.special import factorial
result = factorial(5, exact=True)
print(result)

A 120

B 24

C 720

D Error

💡 Hint

Recall that factorial of 5 is 5*4*3*2*1.

The factorial of 5 is 120 because 5*4*3*2*1 equals 120. Using exact=True returns an integer.

❓ data_output

intermediate

Gamma function values for half-integers

What is the output array from this code using scipy's gamma function for half-integers?

SciPy

import numpy as np
from scipy.special import gamma
x = np.array([0.5, 1.5, 2.5])
result = gamma(x)
print(result)

A [0.88622693 1.32934039 3.32335097]

B TypeError

C [1.0 1.0 1.0]

D [1.77245385 0.88622693 1.32934039]

💡 Hint

Gamma(0.5) equals sqrt(pi).

Gamma(0.5) is sqrt(pi) ≈ 1.772, Gamma(1.5) = 0.5*sqrt(pi) ≈ 0.886, Gamma(2.5) = 1.5*0.5*sqrt(pi) ≈ 1.329.

❓ visualization

advanced

Plotting factorial vs gamma function

Which option produces the correct plot comparing factorial and gamma functions for integers 1 to 6?

SciPy

import matplotlib.pyplot as plt
from scipy.special import factorial, gamma
import numpy as np
x = np.arange(1,7)
plt.plot(x, factorial(x, exact=True), 'o-', label='factorial')
plt.plot(x, gamma(x+1), 'x--', label='gamma(x+1)')
plt.legend()
plt.show()

A Plot shows two overlapping lines with factorial and gamma(x+1) values identical for integers 1 to 6.

B Plot shows factorial values increasing but gamma values constant at 1.

C Plot shows factorial values decreasing and gamma values increasing.

D Plot raises a TypeError due to incompatible types.

💡 Hint

Recall gamma(n+1) = n! for positive integers n.

Gamma function shifted by 1 matches factorial exactly for positive integers, so plots overlap.

🧠 Conceptual

advanced

Understanding gamma function domain

Which statement about the gamma function domain is correct?

A Gamma function is defined for all real numbers including zero.

B Gamma function is defined for all real numbers except non-positive integers.

C Gamma function is only defined for positive integers.

D Gamma function is only defined for complex numbers with positive real part.

💡 Hint

Think about poles of the gamma function.

Gamma function has poles at zero and negative integers, so it is undefined there but defined elsewhere.

🔧 Debug

expert

Identify error in gamma function usage

What error does this code raise?

SciPy

from scipy.special import gamma
result = gamma(-2)
print(result)

A ValueError: math domain error

B ZeroDivisionError

C No error, prints inf

D OverflowError

💡 Hint

Gamma has poles at negative integers, check what scipy returns.

Gamma at negative integers returns infinity (inf) in scipy, no exception is raised.