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ntruwant
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Joined: August 3rd, 2004, 9:50 am

complex numbers

June 20th, 2007, 5:31 am

Hicould anybody help me with exp(ix) where i²=-1. Can you tell that it is 0 at x=+infinity and x=-infinity and why?thanks!
 
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Zedr0n
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complex numbers

June 20th, 2007, 6:13 am

Hm, exp(ix) is a circle as exp(ix) = cosx + isinx so it isn't really defined at infinity
 
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ntruwant
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complex numbers

June 20th, 2007, 8:07 am

Thanks Zedr0n!It's for the Fourier transform that is used in the Heston model. It transforms differentiation into multiplication.When you integrate by parts, you obtain a term that is f(x) * exp(ix y) and this term vanishesat + en - infinity.Does anybody know why this term vanishes?
 
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BlackSheep
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complex numbers

June 20th, 2007, 8:26 am

Have a look at Riemann-Lebesgue's lemma.
 
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ntruwant
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complex numbers

June 20th, 2007, 10:12 am

thanks blacksheep!When I google this, I see it has something to do with the convergence of the Fourier series, but I don't get the lemma.Can you say this term dissapears because of the Riemann-Lebesgue's lemma?
 
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ppauper
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complex numbers

June 20th, 2007, 12:26 pm

QuoteOriginally posted by: ntruwantHicould anybody help me with exp(ix) where i²=-1. Can you tell that it is 0 at x=+infinity and x=-infinity and why?since we're dealing with complex numbers, let's write x=x_r + i x_i where x_r and x_i are the real and imaginary parts of xexp(i x) =exp(i x_r) exp(i^2 x_i) =(cos x_r + i sin x_r) exp(-x_i) x -> "infinity"if the imaginary part of x is > 0, exp(ix) will vanish as x -> "infinity"
 
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fizik
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complex numbers

June 20th, 2007, 7:49 pm

QuoteWhen you integrate by parts, you obtain a term that is f(x) * exp(ix y) and this term vanishesat + en - infinity.I guess, it has more to do with the properties of f(x). It is probably constrained to vanish at x = +- infinity. At least it must vanish there for Riemann-Lebesgue lemma to apply.
 
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ntruwant
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complex numbers

June 21st, 2007, 5:44 am

thanks ppauper!
 
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smo
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complex numbers

June 25th, 2007, 1:17 am

QuoteOriginally posted by: ntruwantHicould anybody help me with exp(ix) where i²=-1. Can you tell that it is 0 at x=+infinity and x=-infinity and why?thanks!limit of exp(ix) as x-> +/- infinity doesn't converge, so it's difficult to assign a meaning to it.