The following distributional limit is well-known: <munder> <mo movablelimits="true" form="pr

Simone Werner 2022-05-28 Answered
The following distributional limit is well-known:
lim a 0 + ( 1 x i a 1 x + i a ) = 2 i π δ ( x ) .
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Answers (1)

barbesdestyle2k
Answered 2022-05-29 Author has 10 answers
Define f α ( y ) := 1 ( y i ) α 1 ( y + i ) α , I α := R f α ( y ) d y so distributionally
lim a 0 + a α 1 ( 1 ( x i a ) α 1 ( x + i a ) α ) = lim a 0 + 1 a f α ( x a ) = I α δ ( x ) .
In particular, I α = R ( y + i ) α ( y i ) α ( y 2 + 1 ) α d y is finite for α ( 0 , 1 ), because its integrand is asymptotic to 2 i α y α 1 | y | . But you've asked about what we get by multiplying this by a 1 α , which for α ( 0 , 1 ) gives 0.
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