Explanation Approach: The solution to the Bessel equation of first kind of order v is given as, J v ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v + n ) ( x 2 ) 2 n + v . Γ ( n + 1 ) = n Γ ( n ) . n ! = n ( n − 1 ) ( n − 2 ) ... ( 3 ) ( 2 ) ( 1 ) . Calculation: Consider the solution to the Bessel equation of first kind of order v . J v ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v + n ) ( x 2 ) 2 n + v And for the order ( v + 1 ) the solution is, J v + 1 ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 2 + v + n ) ( x 2 ) 2 n + v + 1 ... ( 1 ) Differentiate the above equation with respect to x , this gives, J ′ v ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v + n ) ( 2 n + v 2 ) ( x 2 ) 2 n + v − 1 2 J ′ v ( x ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + v ) n ! Γ ( 1 + v + n ) ( x 2 ) 2 n + v − 1 For the order ( v − 1 ) , the solution becomes, J v − 1 ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( 1 + v − 1 + n ) ( x 2 ) 2 n + v − 1 = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( v + n ) ( x 2 ) 2 n + v − 1 Evaluate the expression [ J v − 1 ( x ) − 2 J ′ v ( x ) ] as follows, J v − 1 ( x ) − 2 J ′ v ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( v + n ) ( x 2 ) 2 n + v − 1 − ∑ n = 0 ∞ ( − 1 ) n ( 2 n + v ) n ! Γ ( 1 + v + n ) ( x 2 ) 2 n + v − 1 = ∑ n = 0 ∞ ( − 1 ) n n ! Γ ( v + n ) ( x 2 ) 2 n + v − 1 − ∑ n = 0

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Chapter 8.8, Problem 24E

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The recurrence formula (34).

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Chapter 8.8, Problem 23E

Chapter 8.8, Problem 25E

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To derive: The recurrence formula (34) .

Solution Summary: The author explains how the recurrence formula (34) is derived from the solution to the Bessel equation of first kind of order v.

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To derive:

The recurrence formula (34).

Want to see the full answer?

Chapter 8 Solutions

To derive: The recurrence formula (34) .

Solution Summary: The author explains how the recurrence formula (34) is derived from the solution to the Bessel equation of first kind of order v.

Videos

To derive: The recurrence formula (34).

Want to see the full answer?

Chapter 8 Solutions

To derive:

The recurrence formula (34).