Explanation The Bessel function is represented as follows: J v ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v + n ) ( x 2 ) 2 n + v The recurrence relation is represented as follows: J v ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v + n ) ( x 2 ) 2 n + v ...... ( 1 ) Substitute − 1 2 for v in equation (1). J − 1 2 ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 − 1 2 + n ) ( x 2 ) 2 n − 1 2 ...... ( 2 ) For n = 1 the gamma function can be written as shown below

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 6.4, Problem 31E

To determine

The derivation of elementary form of J−12(x).

Expert Solution & Answer

Chapter 6 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - In Problems 110 find the interval and radius of...Ch. 6.1 - Prob. 10E

Ch. 6.1 - Prob. 11E Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - Prob. 13E Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - In Problems 1116 use an appropriate series in (2)...Ch. 6.1 - Prob. 16E Ch. 6.1 - In Problems 17 and 18 use an appropriate series in...Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - Prob. 28E Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 2530 proceed as in Example 3 to...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - In Problems 3134 verify by direct substitution...Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - In Problems 3538 proceed as in Example 4 and find...Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.2 - In Problems 1 and 2 without actually solving the...Ch. 6.2 - Prob. 2E Ch. 6.2 - In Problems 3–6 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - In Problems 36 find two power series solutions of...Ch. 6.2 - Prob. 7E Ch. 6.2 - Prob. 8E Ch. 6.2 - Prob. 9E Ch. 6.2 - Prob. 10E Ch. 6.2 - Prob. 11E Ch. 6.2 - Prob. 12E Ch. 6.2 - Prob. 13E Ch. 6.2 - Prob. 14E Ch. 6.2 - Prob. 15E Ch. 6.2 - Prob. 16E Ch. 6.2 - Prob. 17E Ch. 6.2 - Prob. 18E Ch. 6.2 - Prob. 19E Ch. 6.2 - Prob. 20E Ch. 6.2 - Prob. 21E Ch. 6.2 - Prob. 22E Ch. 6.2 - Prob. 23E Ch. 6.2 - In Problems 23 and 24 use the procedure in Example...Ch. 6.2 - Without actually solving the differential equation...Ch. 6.2 - How can the power series method be used to solve...Ch. 6.2 - Prob. 27E Ch. 6.2 - Prob. 28E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 3E Ch. 6.3 - Prob. 4E Ch. 6.3 - Prob. 5E Ch. 6.3 - Prob. 6E Ch. 6.3 - Prob. 7E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 9E Ch. 6.3 - In Problems 110 determine the singular points of...Ch. 6.3 - Prob. 11E Ch. 6.3 - Prob. 12E Ch. 6.3 - In Problems 13 and 14, x = 0 is a regular singular...Ch. 6.3 - Prob. 14E Ch. 6.3 - In Problems 1524, x = 0 is a regular singular...Ch. 6.3 - Prob. 16E Ch. 6.3 - Prob. 17E Ch. 6.3 - Prob. 18E Ch. 6.3 - Prob. 19E Ch. 6.3 - Prob. 20E Ch. 6.3 - Prob. 21E Ch. 6.3 - Prob. 22E Ch. 6.3 - Prob. 23E Ch. 6.3 - Prob. 24E Ch. 6.3 - Prob. 25E Ch. 6.3 - Prob. 26E Ch. 6.3 - Prob. 27E Ch. 6.3 - Prob. 28E Ch. 6.3 - Prob. 29E Ch. 6.3 - Prob. 30E Ch. 6.3 - Prob. 31E Ch. 6.3 - Prob. 32E Ch. 6.3 - Prob. 33E Ch. 6.3 - Prob. 35E Ch. 6.3 - Prob. 36E Ch. 6.3 - Prob. 37E Ch. 6.4 - Prob. 1E Ch. 6.4 - Prob. 2E Ch. 6.4 - Prob. 3E Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Bessels Equation In Problems 16 use (1) to find...Ch. 6.4 - Prob. 7E Ch. 6.4 - Prob. 8E Ch. 6.4 - Prob. 9E Ch. 6.4 - Prob. 10E Ch. 6.4 - In Problems 11 and 12 use the indicated change of...Ch. 6.4 - In Problems 11 and 12 use the indicated change of...Ch. 6.4 - Prob. 13E Ch. 6.4 - Prob. 14E Ch. 6.4 - Prob. 15E Ch. 6.4 - Prob. 16E Ch. 6.4 - Prob. 17E Ch. 6.4 - Prob. 18E Ch. 6.4 - Prob. 19E Ch. 6.4 - In Problems 1320 use (20) to find the general...Ch. 6.4 - Prob. 21E Ch. 6.4 - Assume that b in equation (20) can be pure...Ch. 6.4 - Prob. 23E Ch. 6.4 - Prob. 24E Ch. 6.4 - In Problems 2326 first use (20) to express the...Ch. 6.4 - In Problems 2326 first use (20) to express the...Ch. 6.4 - Prob. 27E Ch. 6.4 - Prob. 28E Ch. 6.4 - Prob. 29E Ch. 6.4 - Prob. 30E Ch. 6.4 - Prob. 31E Ch. 6.4 - Use the recurrence relation in Problem 28 along...Ch. 6.4 - Prob. 33E Ch. 6.4 - Prob. 34E Ch. 6.4 - Use the change of variables s=2kmet/2 to show that...Ch. 6.4 - Prob. 36E Ch. 6.4 - Use the result in parts (a) and (b) of Problem 36...Ch. 6.4 - Prob. 38E Ch. 6.4 - Prob. 39E Ch. 6.4 - (a) Use the explicit solutions y1(x) and y2(x) of...Ch. 6.4 - Prob. 47E Ch. 6.4 - Prob. 48E Ch. 6.4 - Find the first three positive values of for which...Ch. 6.4 - The differential equation y 2xy + 2y = 0 is known...Ch. 6.4 - (a) When = n is a nonnegative integer, Hermites...Ch. 6.4 - Prob. 55E Ch. 6.4 - Prob. 56E Ch. 6 - In Problems 1 and 2 answer true or false without...Ch. 6 - Prob. 2RE Ch. 6 - Prob. 3RE Ch. 6 - Prob. 4RE Ch. 6 - Prob. 5RE Ch. 6 - Prob. 6RE Ch. 6 - Prob. 7RE Ch. 6 - Prob. 8RE Ch. 6 - Prob. 9RE Ch. 6 - Prob. 10RE Ch. 6 - Prob. 11RE Ch. 6 - Prob. 12RE Ch. 6 - Prob. 13RE Ch. 6 - Prob. 14RE Ch. 6 - Prob. 15RE Ch. 6 - Prob. 16RE Ch. 6 - Prob. 17RE Ch. 6 - Prob. 18RE Ch. 6 - Prob. 19RE Ch. 6 - Prob. 20RE Ch. 6 - Prob. 21RE Ch. 6 - Prob. 22RE Ch. 6 - Prob. 23RE Ch. 6 - Prob. 24RE Ch. 6 - Prob. 25RE Ch. 6 - Express the general solution of the given...Ch. 6 - Prob. 27RE Ch. 6 - Prob. 28RE

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage