- #1

- 10

- 0

let

assuming x = 2*pi*v*t, why is E(t) multiplied by e^(-ix)?, i guess it has to do with the fact that it is the conjugate of e^(ix), but i can't figure it out

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In summary, the first step of the Plancherel's Theorem proof involves using the definition of Fourier Transform pairs and taking the complex conjugate of the first equation. This is done by changing the variable in the integral and using the fact that definite integrals are unaffected by variable changes.

- #1

- 10

- 0

let

assuming x = 2*pi*v*t, why is E(t) multiplied by e^(-ix)?, i guess it has to do with the fact that it is the conjugate of e^(ix), but i can't figure it out

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- #2

Gold Member

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The second equation follows from first by taking the complex conjugate at each side of the first equation. And of course changing the name of the variable but I guess you know that ##\int f(x)dx=\int f(y)dy## no matter what x and y are.

- #3

- 10

- 0

f(x) = x³

y = x²

then

f(y) = (x²)³ = x⁶

dy/dx = 2x

dy = 2x dx

using the equation you suggest:

∫f(x)dx=∫f(y)dy

∫x³ dx=∫2x⁷ dx

i'm missing something?

- #4

Gold Member

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- 2,625

What you doing is a change of variable ##y=x^2## in the integral ##\int\limits_{a}^{b}f(y)dy## so the interval of integration changes from ##(a,b)## to ## (\sqrt{a},\sqrt{b})##. So the last line of your post should be actually ##\int\limits_{a}^{b}x^3dx=\int\limits_{\sqrt{a}}^{\sqrt{b}}2x^7dx##.

Plancherel's Theorem is a mathematical theorem that relates the Fourier transform of a function to its square-integral. It is named after French mathematician Michel Plancherel.

Plancherel's Theorem states that the integral of the squared absolute value of a function in one domain is equal to the integral of the squared absolute value of its Fourier transform in the other domain.

Plancherel's Theorem is a fundamental result in Fourier analysis and is used in various fields of mathematics, physics, and engineering. It allows for the conversion between functions in the time and frequency domains, making it a powerful tool for studying and analyzing signals and systems.

The proof of Plancherel's Theorem involves using the properties of the Fourier transform, such as linearity, duality, and the convolution theorem. It also uses techniques from measure theory and complex analysis. The complete proof can be found in most advanced mathematical analysis textbooks.

Yes, there are variations of Plancherel's Theorem for different types of Fourier transforms, such as the discrete-time Fourier transform and the discrete Fourier transform. These variations have similar principles but are tailored to different types of signals and systems.

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