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Bessel functions Bessel functions arise in the study of
Chapter 8, Problem 75E(choose chapter or problem)
Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
\(J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2^{2 k}(k !)^{2}} x^{2 k}\).
a. Write out the first four terms of \(J_{0}\).
b. Find the radius and interval of convergence of the power series for \(J_{0}\).
c. Differentiate \(J_{0}\), twice and show (by keeping terms through \(x^{6}\)) that \(J_{0}\) satisfies the equation \(x^{2} y^{\prime \prime}(x)+x y^{\prime}(x)+x^{2} y(x)=0\).
Questions & Answers
QUESTION:
Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
\(J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2^{2 k}(k !)^{2}} x^{2 k}\).
a. Write out the first four terms of \(J_{0}\).
b. Find the radius and interval of convergence of the power series for \(J_{0}\).
c. Differentiate \(J_{0}\), twice and show (by keeping terms through \(x^{6}\)) that \(J_{0}\) satisfies the equation \(x^{2} y^{\prime \prime}(x)+x y^{\prime}(x)+x^{2} y(x)=0\).
ANSWER:Solution 75AE
Step 1:
In this problem we have to find the first four terms of
a. Write out the first four terms of J0.
Consider
We need to find the first four terms.
Put k = 0
We get, (since
)
Therefore first term of is 1
Put k = 1
We get,
Therefore second term of is
Put k = 2
We get,
Therefore third term of is
Put k = 3
We get,
Therefore fourth term of is