Attain a two-term approximation for the following integral as m goes to 1 from below:

rjawbreakerca 2022-07-07 Answered
Attain a two-term approximation for the following integral as m goes to 1 from below:
I = 0 π / 2 d θ 1 ( m 2 ) sin ( θ ) 2 .
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Answers (1)

Mekjulleymg
Answered 2022-07-08 Author has 14 answers
This is just the complete elliptic intagral of the first kind, thus
I = K ( m ) = 1 2 π n = 0 Γ 2 ( n + 1 2 ) Γ 2 ( n + 1 ) ( 1 m 2 ) n ( log ( 1 m 2 ) + a n )
as m 1 0, where a 0 = 2 log 2 and
a n + 1 = a n + 4 ( 2 n + 1 ) ( 2 n + 2 ) ( n 0 ) .
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