Chapter 12: Q6P (page 590)

To show the following equation shown in the problem
$2 \frac{d}{d x} J_{a} (x) = J_{a - 1} (x)$ .

Short Answer

Expert verified

The answer is, $2 \frac{d}{dx} J_{a} (x) = J_{a - 1} (x)$ .

Step by step solution

Step 1: Concept of Bessel Function:

The Bessel functions of the first kind $J_{n} (x)$ are defined as the solutions to the Bessel differential equation.

$x^{2} \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + (x^{2} - n^{2}) y = 0$

Solve the equation 2ddxJa(x)=Ja-1(x)-Ja+1(x) about the Bessel Function:

The given equation for the following equation is shown as,

$2 \frac{d}{dx} J_{a} (x) = J_{a - 1} (x) - J_{a + 1} (x)$

By the use of the following proposition about the zeros of Bessel functions solve the equation as follows:

\frac{d}{d x} J_{α} (x) = \frac{d}{d x} (x^{a} (x^{- α} J_{α} (x))) = α x^{α - 1} (J_{α} (x)) + x^{α} \frac{d}{d x} (x^{α} J_{α} (x)) = - α x^{α - 1} J_{a} (x) + x^{- α} (x^{α} J_{α - 1} (x)) = - α x^{α - 1} J_{α} (x) + J_{a - 1} (x)

Calculation of the equation ddx(xαx-αJα(x) :

By adding the two expressions as follows:

$\frac{d}{d x} J_{a} (x) = \frac{d}{d x} (x^{α} (x^{- α} J_{α}) (x)) = α x^{α - 1} (J_{α} (x)) + x^{α} \frac{d}{d x} (x^{α} J_{α} (x)) = (J_{α - 1} (x) - J_{α + 1} (x)) / 2 2 \frac{d}{d x} J_{α} (x) = J_{a - 1} (x) - J_{a + 1} (x)$

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To show the following equation shown in the problem
$2 \frac{d}{d x} J_{a} (x) = J_{a - 1} (x)$ .

Short Answer

Step by step solution

Step 1: Concept of Bessel Function:

Solve the equation 2ddxJa(x)=Ja-1(x)-Ja+1(x) about the Bessel Function:

Calculation of the equation ddx(xαx-αJα(x) :

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To show the following equation shown in the problem2ddxJa(x)=Ja-1(x).

Short Answer

Step by step solution

Step 1: Concept of Bessel Function:

Solve the equation 2ddxJa(x)=Ja-1(x)-Ja+1(x) about the Bessel Function:

Calculation of the equation ddx(xαx-αJα(x) :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Thermodynamics

Engineering Physics

Scientific Method Physics

Dynamics

Measurements

Study anywhere. Anytime. Across all devices.

To show the following equation shown in the problem
$2 \frac{d}{d x} J_{a} (x) = J_{a - 1} (x)$ .