improper integral: exp(ikx) undefined in Matlab?
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Steven Lord on 19 Jun 2021
What is the value of your integral when k is equal to 0?
syms x k;
k = 0;
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;
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--')
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);
So should the value of this integral on the infinite interval be 0, -2, or something inbetween?