How to find lim x → − N J a = ( − 1 ) N J N

Use the fact that the reciprocal of the Euler's Γ-function vanishes at non-positive integers, that is for k ∈ Z ⩽ 0 lim x → − k 1 Γ ( x ) = 0Thus, all the terms where 0 ⩽ k &lt; N vanish in the limit: lim a → − N J a ( x ) = ∑ k = N ∞ ( x 2 ) 2 k − N ( − 1 ) k k ! ⋅ Γ ( 1 − N + k ) = ∑ k = 0 ∞ ( x 2 ) 2 k + N ( − 1 ) k + N Γ ( k + N + 1 ) ⋅ k ! = ( − 1 ) N J N ( x )

Solved: "How to find lim x → − N..."

sondestiny120g

Answered question

2022-08-11

How to find

lim_{x \to - N} J_{a} = (- 1)^{N} J_{N}

Answer & Explanation

Dominic Paul

Beginner2022-08-12Added 17 answers

Use the fact that the reciprocal of the Euler's

Γ

-function vanishes at non-positive integers, that is for

k \in Z_{⩽ 0}

lim_{x \to - k} \frac{1}{Γ (x)} = 0

Thus, all the terms where

0 ⩽ k < N

vanish in the limit:

\begin{array}{rcl} lim_{a \to - N} J_{a} (x) & = & \sum_{k = N}^{\infty} {(\frac{x}{2})}^{2 k - N} \frac{(- 1)^{k}}{k! \cdot Γ (1 - N + k)} = \sum_{k = 0}^{\infty} {(\frac{x}{2})}^{2 k + N} \frac{(- 1)^{k + N}}{Γ (k + N + 1) \cdot k!} \\ = & (- 1)^{N} J_{N} (x) \end{array}

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