Connect the curve 4

Calculus Level 5

The figure shows the curves of \(y=\cos { \left( \cos { \left( x \right) } \right) } \) in blue and \(y=\sin { \left( \sin { \left( x \right) } \right) } \) in red.

It also shows a green region enclosed between these graphs in domain \(\left[ 0,2\pi \right] \).

If the area of the green region is \[A{ \pi }^{ B }{ J }_{ \alpha }\left( n \right) \]

Find \(A+B+\alpha+n\)

Details and Assumptions

  1. \({ J }_{ \alpha }\left( n \right)\) is a Bessel function of the first kind.

Or \[J_{ \alpha }(n)=\sum _{ m=0 }^{ \infty } \frac { (-1)^{ m } }{ m!\, \Gamma (m+\alpha +1) } { \left( \frac { n}{ 2 } \right) }^{ 2m+\alpha }\]

  1. \(A\), \(B\), \(\alpha\) and \(n\) are integers.

  2. \(\alpha<n\)

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