Find the leading behavior of <msubsup> &#x222B;<!-- ∫ --> 0 <mi mathvariant="norm

Emanuel Keith

Emanuel Keith

Answered question

2022-06-27

Find the leading behavior of 0 cos ( x ( t 3 3 t ) ) d t

Answer & Explanation

massetereqe

massetereqe

Beginner2022-06-28Added 21 answers

I = 0 cos ( x ( t 3 3 t ) ) d t
Knowing the properties of the Airy functions, the answer is easy :
Let { T 3 = x t 3 z T = x t z = x 2 / 3
I = x 1 / 3 0 cos ( T 3 3 z T ) d T = x 1 / 3 π Ai ( z )
Ai is the Airy function :
I = π x 1 / 3 Ai ( x 2 / 3 )
Equivalent at infinity :
Ai ( z ) 1 π z 1 / 4 sin ( 2 3 z 3 / 2 + π 4 ) + O ( 1 z 3 / 2 )
I = 0 cos ( x ( t 3 3 t ) ) d t π x sin ( 2 3 x + π 4 ) + O ( 1 x )

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?