Fourier Transform Example

Fourier Transform Example

Fourier transforms are hard!! Which is why it would be nice to have a concrete example of a Fourier transform, which we do precisely in this video. Here we calculate the Fourier transform of the Gaussian function e^-x^2 or exp(-x squared) and the beautiful result is that it is again a Gaussian! This self-similarity or fractal property will be crucial when we'll solve the heat equation using the Fourier transform

[rommelpamplona]

Good day I liked the exercise in this particular issue with my colleagues engineers we start watching their videos and they are very interesting and we learn a lot, thanks for teaching us and greetings from South America !!!

[Mathguy]

Considering a∈ℝ, ∫ℝ exp(-𝓍²-2a𝓍) d𝓍 = ∫ℝ exp(-(𝓍+a)² + a²) d𝓍 = exp(a²)√π. By analytic continuation, this formula works for a∈ℂ. Let a = 𝓲k, we get the desired fourier transform.

[explsn]

Other than using ode, we can also solve it by Cauchy-Goursat’s theorem (yes! No need residue formula!)

[Sam-vnmh]

I like Fourier transform of a function by using hilbert space and creating it by vector spaces

[syncshot]

I love you man, these videos are great!

[jellyfrancis]

"Usually written in XI but I can't write XI" been there man....some letter I can't write ever xd

[lazarussevy]

Your integral sign looks really nice :)

[pierreabbat]

What's the Hadamard transform of a Gaussian?

[theproofessayist]

for the integrand: xe^(-x^2)*e^(ikx) we cannot combine the exponents together into (-x^2+ikx) and use a U substitution there because we would get a 2x+ik (the 2x is nice but we can do nothing about the ik right) hence why doing the way you are doing?

[alipourzand]

The heat is on ...
Song for the next video ☺

[gosufd]

Thanks very much. This is also obtained from infinite product of sinh
Prod [(1+x^2/k^2)] = sinh (pi x) / (pi x)
Taking log
Sum log [1+x^2/k^2] = log (sinh x) - log (pi x)
Differentiate by x
Sum 2x/(x^2 + k^2) = pi coth (pi x) - 1/x
set x = 1

[renesperb]

The word ''any'' in the title is a little exagerated . You can solve some types of PDE's
the way you do it here.But they have to be linear and of a special simple form.

[georgekibunja]

Nice job, express a complex number z=(i+√2)⅓ in polar form, kindly

[Man]

I have two queries..i don't have a proof if these two are correct.
(1)Let "a1" & "a2" & "a3" ...."an" be integers such that all of them are non zero .Is lcm(a1, a2, a3, ..., an)=lcm(|a1|, |a2|, |a3|, ..., |an|)?

(2)Let "a1" & "a2" & "a3" ...."an" be integers such that atleast one of them is non zero .Is gcd(a1, a2, a3, ..., an)=gcd(|a1|, |a2|, |a3|, ..., |an|)?.

[SarojtapashLalita]

Why you write everything in capital letters

[FleuveAlphee]

The ODE is separable, so we don't need an integrating factor. Nice job otherwise.

[rob]

d/dk f = -k/2 f
=> ∫df/f = -1/2 ∫ k dk
ln f = -k^2/4 + c
f = e^(-k^2/4 + c) = C e^(-k^2/4)