Explanation Given: The power series expansion for J v ( λ x ) has the form ( λ x ) v g ( λ x ) , where g is analytic about x = 0 . Approach: For an infinite power series expansion, the Bessel’s function of the first kind of order v is given by, J v ( x ) = ∑ n = 0 ∞ ( − 1 ) n ( x 2 ) v + 2 n n ! ( v + n ) ! ... ( 1 ) Replace x by λ x in the above equation and then by differentiating the obtained result get the desired result. Calculation: Replace x by λ x in the equation ( 1 ) , this implies, J v ( λ x ) = ∑ n = 0 ∞ ( − 1 ) n ( λ x 2 ) v + 2 n n ! ( v + n ) ! = ∑ n = 0 ∞ ( − 1 ) n ( λ x 2 ) v ( λ x 2 ) 2 n n ! ( v + n ) ! = ( λ x ) v ∑ n = 0 ∞ ( − 1 ) n ( 1 2 ) v ( λ x 2 ) 2 n n ! ( v + n ) ! … ( 2 ) Suppose, g ( λ x ) = ∑ n = 0 ∞ ( − 1 ) n ( 1 2 ) v ( λ x 2 ) 2 n n ! ( v + n ) ! .... ( 3 ) Thus, J v ( λ x ) = ( λ x ) v g ( λ x ) . Take the limit in the equation ( 2 ) on both sides,

Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

7th Edition

ISBN: 9780134768717

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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Chapter 11.7, Problem 9E

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That for the order v=0 or v≥1, both v≥1, both Jv(λx) and J′v(λx) remain bounded as x→0+.

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Chapter 11.7, Problem 8E

Chapter 11.7, Problem 10E

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Chapter 11 Solutions

Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

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That for the order v = 0 or v ≥ 1 , both v ≥ 1 , both J v ( λ x ) and J ′ v ( λ x ) remain bounded as x → 0 + .

Solution Summary: The author illustrates how the Bessel's function of the first kind of order v is given by J_v(sqrtlambda x

Videos

To show:

That for the order v=0 or v≥1, both v≥1, both Jv(λx) and J′v(λx) remain bounded as x→0+.

Want to see the full answer?

Chapter 11 Solutions

That for the order v = 0 or v ≥ 1 , both v ≥ 1 , both J v ( λ x ) and J ′ v ( λ x ) remain bounded as x → 0 + .

Solution Summary: The author illustrates how the Bessel's function of the first kind of order v is given by J_v(sqrtlambda x

Videos

To show: That for the order v=0 or v≥1, both v≥1, both Jv(λx) and J′v(λx) remain bounded as x→0+.

Want to see the full answer?

Chapter 11 Solutions

To show:

That for the order v=0 or v≥1, both v≥1, both Jv(λx) and J′v(λx) remain bounded as x→0+.