Question

Given the Bessel function:
$$ J_0(x) = 1 - \frac{x^2}{2^2} + \frac{x^4}{2^2 \cdot 2^2} - \frac{x^6}{2^2 \cdot 2^2 \cdot 2^2} + \dots $$ The Bessel function $ J_1(x) $ is given by:

• A. \( \frac{x}{2} - \frac{x^3}{2^2 \cdot 4} + \frac{x^5}{2^2 \cdot 4 \cdot 6} - \dots \)
• B. \( 1 + \frac{x^2}{2^2} - \frac{x^4}{2^2 \cdot 4^2} + \dots \)
• C. \( \frac{x}{2} - \frac{x^3}{2^2 \cdot 2^2} + \frac{x^5}{2^2 \cdot 2^2 \cdot 6} - \dots \)
• D. None of the three

Hint:

Bessel functions appear in wave equations in cylindrical or spherical coordinates.

Answer 1

The Correct Option is A

The Bessel function of the first kind, \( J_1(x) \), is given by:
\[ J_1(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k! \Gamma(k+2)} \left( \frac{x}{2} \right)^{2k+1} \] which simplifies to:
\[ J_1(x) = \frac{x}{2} - \frac{x^3}{2^2 \cdot 4} + \frac{x^5}{2^2 \cdot 4 \cdot 6} - \dots \]