Dealing with complex integrals of a highly oscillatory type

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xadu 2020-7-29

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评论： xadu 2020-7-29

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The following integral can be computed analytically using complex integration (required due to the pole at the origin):

However, if I try to evaluate this numerically, I run into trouble. Two approaches

1. Numerically: This is doomed to fail due to the function's behaviour at the origin.

fun = @(x) exp(1j*x)./x;
integral(fun,-inf,inf)
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error
is   9.8e+00. The integral may not exist, or it may be difficult to approximate numerically to
the requested accuracy. 

2. Symbolically: This gives the wrong answer and leaves a limit to be evaluated

syms x;
f(x) = exp(1j*x)/x;
int(f,[-inf,inf],'PrincipalValue',true)
ans =
- pi*1i + limit(- ei(-1i/eps27) + ei(1i/eps27), eps27, 0, 'Right')

What is the way to evaluating such integrals in matlab? I suspect the trouble is not at the origin, but at how the function is behaving (numerically) at the end points.

PS: I did try to use the chebfun library here, but it didn't seem to help.

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Answer 1

Alan Stevens 2020-7-29

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I don't know in general, but for this particular integral, since cos(x)/x is odd, its contribution to the integral is zero and we can do the following:

>> syms x
>> I = i*int(sin(x)/x,-inf,inf)
 
I =
 
pi*1i

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xadu 2020-7-29

True, for this particular case it solves the problem.. but the problem remains if a general oscillatory type of integral is encountered. In fact, I came across this problem while working on a (complicated looking) kernel -- and then boiled it down to the above, simpler integral for the purpose of posting the question. Thanks, though :).

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