evaluate-the-following-integral-in-closed-form-or-express-it-in-terms-of-known-special-functions-K-i-at-J-bt-1-dt-where-K-i-z-is-the-modified-Bess

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 .