i generalized Bessel function′s Laplace Transform L.TJ_ν (z)=(((s+(√(s^2 +1)))^(−ν) )/( (√(s^2 +1)))) , s∈[0,∞) , ν∈R L.T Y_ν (z)=((cot(πν)(s+(√(s^2 +1)))^(−ν) )/( (√(s^2 +1))))−((csc(πν)(s+(√(s^2 +1)))^ν )/( (√(s^2 +1)))) s∈[0,∞) , ν∈R^+ \{0,Z^+ } but.... i can′t explain why L.T Y_ν (z) is undefined when ν∈Z^+ \{0}..... Help me.....!!!


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Question Number 212765 by issac last updated on 23/Oct/24
$i generalized Bessel function′s Laplace Transform L.TJ_ν (z)=(((s+(√(s^2 +1)))^(−ν) )/( (√(s^2 +1)))) , s∈[0,∞) , ν∈R L.T Y_ν (z)=((cot(πν)(s+(√(s^2 +1)))^(−ν) )/( (√(s^2 +1))))−((csc(πν)(s+(√(s^2 +1)))^ν )/( (√(s^2 +1)))) s∈[0,∞) , ν∈R^+ \{0,Z^+ } but.... i can′t explain why L.T Y_ν (z) is undefined when ν∈Z^+ \{0}..... Help me.....!!!$
$i generalized Bessel {function}^{'} s$ $Laplace Transform$ $L . T J_{ν} (z) = \frac{{(s + \sqrt{s^{2} + 1})}^{- ν}}{\sqrt{s^{2} + 1}}, s \in [0, \infty), ν \in R$ $L . T Y_{ν} (z) = \frac{\cot (π ν) {(s + \sqrt{s^{2} + 1})}^{- ν}}{\sqrt{s^{2} + 1}} - \frac{\csc (π ν) {(s + \sqrt{s^{2} + 1})}^{ν}}{\sqrt{s^{2} + 1}}$ $s \in [0, \infty), ν \in R^{+} ∖ {0, Z^{+}}$ $but . . . . i {can}^{'} t explain why L . T Y_{ν} (z)$ $is undefined when ν \in Z^{+} ∖ {0} . . . . .$ $Help me . . . . .!!!$
	Answered by Frix last updated on 23/Oct/24

	$\cot π ν = \frac{1}{\tan π ν} \Rightarrow \tan π ν \neq 0 \Rightarrow ν \neq n \forall n \in Z$ $\csc π ν = \frac{1}{\sin π ν} \Rightarrow \sin π ν \neq 0 \Rightarrow ν \neq n \forall n \in Z$
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