Bessel functions in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Bessel function call

What is the output of this Python code using SciPy's Bessel function?

SciPy

from scipy.special import jv
result = jv(2, 3.0)
print(round(result, 4))

A 0.3391

B 0.5767

C 0.4861

D 0.2239

💡 Hint

Recall that jv(n, x) computes the Bessel function of the first kind of order n at x.

The function jv(2, 3.0) returns approximately 0.3391. This is the Bessel function of order 2 evaluated at 3.0.

❓ data_output

intermediate

Number of zeros of Bessel function in range

How many zeros does the Bessel function of the first kind of order 0 have between 0 and 20?

SciPy

from scipy.special import jn_zeros
zeros = jn_zeros(0, 6)
print(len(zeros))

A 7

B 6

C 8

D 5

💡 Hint

The function jn_zeros(n, nt) returns the first nt zeros of the Bessel function of order n.

The first 6 zeros of the Bessel function of order 0 lie between 0 and 20, so the count is 6.

❓ visualization

advanced

Plot Bessel functions of different orders

Which plot correctly shows Bessel functions of the first kind for orders 0, 1, and 2 over the range 0 to 10?

SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import jv
x = np.linspace(0, 10, 100)
plt.plot(x, jv(0, x), label='order 0')
plt.plot(x, jv(1, x), label='order 1')
plt.plot(x, jv(2, x), label='order 2')
plt.legend()
plt.show()

A Three curves starting at 1, 1, and 1 at x=0, constant lines

B Three curves starting at 0, 1, and 2 at x=0, increasing exponentially

C Three curves all starting at 0 at x=0, increasing linearly

D Three curves starting at 1, 0, and 0 respectively at x=0, oscillating and decreasing in amplitude

💡 Hint

Recall that jv(0,0) = 1, jv(n,0) = 0 for n>0.

The Bessel function of order 0 starts at 1 at x=0, while orders 1 and 2 start at 0. All oscillate and their amplitudes decrease as x increases.

🧠 Conceptual

advanced

Behavior of Bessel functions near zero

Which statement correctly describes the behavior of the Bessel function of the first kind jv(n, x) near x=0 for integer order n?

A jv(0, x) approaches 1, and jv(n, x) for n>0 approaches 0 as x approaches 0

B jv(n, x) approaches infinity for all n as x approaches 0

C jv(n, x) oscillates infinitely near zero for all n

D jv(n, x) is undefined at x=0 for all n

💡 Hint

Think about the value of Bessel functions at zero for integer orders.

For integer order n, jv(0, 0) = 1, and jv(n, 0) = 0 for n > 0. They are well-defined and finite at zero.

🔧 Debug

expert

Identify the error in Bessel function usage

What error will this code produce?

SciPy

from scipy.special import jv
result = jv(1.5, 2.0)
print(result)

A ValueError because input must be integer

B TypeError because order must be integer

C Runs correctly and prints a float value

D NameError because jv is not imported

💡 Hint

Check the documentation for jv about allowed order values.

The jv function accepts real-valued orders, so 1.5 is valid and the code runs correctly.