improper integral: exp(ikx) undefined in Matlab?
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Niklas Kurz
le 19 Juin 2021
Modifié(e) : Niklas Kurz
le 20 Juin 2021
I wanna integrate:
With solution:
But Matlab gives NAN:
syms x k; assume(k,'integer'); int(exp(1i*k*x),x,-inf,inf)
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Steven Lord
le 19 Juin 2021
What is the value of your integral when k is equal to 0?
syms x k;
assume(k,'integer');
k = 0;
int(exp(1i*k*x),x,-inf,inf)
This makes sense, as you're just integrating 1 over the whole real line.
What's the value of your integral when k is 1?
k = 1;
int(exp(1i*k*x),x,-inf,inf)
Does this make sense? Let's look at the real and imaginary parts of the function you're integrating.
f = 8*pi;
fplot(real(exp(1i*k*x)), [-f, f], 'k--')
hold on
fplot(imag(exp(1i*k*x)), [-f, f], 'c-')
Those oscillations could be problematic. Does this integral exist? Let's look at a series of values of those integrals for gradually increasing limits.
for L = 0:0.5:10
value = int(exp(1i*k*x), -L*pi, L*pi);
fprintf("The value of the integral from %g*pi to %g*pi is %g.\n", -L, L, value);
end
So should the value of this integral on the infinite interval be 0, -2, or something inbetween?
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