evaluate the following integral in closed form or express it in terms of known special functions; ∫_ ^∞ K_(i𝛌) (at)J_𝛎 (bt)^(𝛍−1) dt where; •K_(i𝛌) (z) is the modified Bessel function of the second kind with imaginary order i𝛌, where 𝛌∈R. • J_𝛎 (z) is the Bessel function of the first kind with order 𝛎 ∈C. • a,b are positif real constants. • 𝛍 is a complex parameter statisfying the conditions for the integral to converge.


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Question Number 218866 by Nicholas666 last updated on 16/Apr/25

$e v a l u a t e t h e f o l l o w i n g i n t e g r a l i n c l o s e d f o r m o r e x p r e s s$ $i t i n t e r m s o f k n o w n s p e c i a l f u n c t i o n s;$ $\int_{}^{\infty} K_{i λ} (a t) J_{ν} {(b t)}^{μ - 1} d t$ $w h e r e;$ $∙ K_{i λ} (z) i s t h e m o d i f i e d B e s s e l f u n c t i o n$ $o f t h e s e c o n d k i n d w i t h i m a g i n a r y o r d e r i λ,$ $w h e r e λ \in R .$ $∙ J_{ν} (z) i s t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d w i t h o r d e r ν \in C .$ $∙ a, b a r e p o s i t i f r e a l c o n s t a n t s .$ $∙ μ i s a c o m p l e x p a r a m e t e r s t a t i s f y i n g t h e$ $c o n d i t i o n s f o r t h e i n t e g r a l t o c o n v e r g e .$
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