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Emanuel Keith 2022-06-27 Answered
Find the leading behavior of 0 cos ( x ( t 3 3 t ) ) d t
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Answers (1)

massetereqe
Answered 2022-06-28 Author has 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 )
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