# Index [[Convergence generating factors]] [[Dirichlet conditions]] [[Existence of Fourier transforms]] [[Fourier convolution theorem]] [[Fourier mode]] [[Fourier representation of 𝛿(y-x)]] [[Fourier series]] [[Fourier transform]] [[Laplace transform]] [[Multi-dimensional Fourier transform]] [[Parseval's identity]] [[Parseval's theorem]] [[Time-frequency Fourier transform]] # Related Indices [Signal Processing and Signal Theory (Index)](Signal%20Processing%20and%20Signal%20Theory%20(Index).md) --- # Proofs and examples [[Proof of the Fourier convolution theorem]] [[Proof of the converse of the Fourier convolution theorem]] [[Proofs of Parseval's theorem]] [Proof that the Fourier transform preserves symmetry](Proof%20that%20the%20Fourier%20transform%20preserves%20symmetry.md) [[Proof that the Fourier transform of the derivative of a function is that of the function itself times an imaginary prefactor]] --- # Basic concepts ## Euler's formula #^abb760) Euler's formula gives us a way to construct [sinusoids](Fourier%20analysis%20(Index).md#Sinusoids) using only [complex exponentials,](Complex%20analysis%20(index).md#Complex%20exponentials) which is convenient for many calculations and derivations in [Fourier analysis.](Fourier%20analysis%20(Index).md) In addition, it allows us to derive every [trigonometric identity](Fourier%20analysis%20(Index).md#Trigonometric%20identities%20of%20Sinusoids) in a convenient manner. ## Periodic functions ### Trigonometric functions #### Sinusoids A sinusoid is a type of trignometric function, $f(t)$ that may be written as $f(t)=A\sin{(2\pi \nu t +\phi)}.$ This includes both [sine and cosine functions](Fourier%20analysis%20(Index).md#Trigonometric%20functions) since #^f35292) #### Trigonometric identities of Sinusoids Given [above](Fourier%20analysis%20(Index).md#Sinusoids) are fundamental algebraic relations between [sine and cosine functions](Fourier%20analysis%20(Index).md#Trigonometric%20functions) and we list additional trigonometric identities that may be derived from these relations below: ##### The Pythagorean Identity $\sin^2(\theta)+\cos^2(\theta)=1$ ##### Half Angle Identities $\sin^2{(\theta)}=\frac{1}{2}(1-\cos{(2\theta)})$ $\cos^2{(\theta)}=\frac{1}{2}(1+\cos{(2\theta)})$ ##### Product to Sum Identities $\sin{(\alpha)}\sin{(\beta)}=\frac{1}{2}[\cos{(\alpha-\beta)}-\cos{(\alpha+\beta)}]$ $\cos{(\alpha)}\cos{(\beta)}=\frac{1}{2}[\cos{(\alpha-\beta)}+\cos{(\alpha+\beta)}]$ $\sin{(\alpha)}\cos{(\beta)}=\frac{1}{2}[\sin{(\alpha+\beta)}+\sin{(\alpha-\beta)}]$ $\cos{(\alpha)}\sin{(\beta)}=\frac{1}{2}[\sin{(\alpha+\beta)}-\sin{(\alpha-\beta)}]$ --- # Bibliography [Altland, A. von Delft, J. _Mathematics for Physicists_, Cambridge University Press, 2019](Altland,%20A.%20von%20Delft,%20J.%20Mathematics%20for%20Physicists,%20Cambridge%20University%20Press,%202019.md) [Brigham E. O., _The Fast Fourier Transform and Its Applications_, Prentice Hall, 1988.](Brigham%20E.%20O.,%20The%20Fast%20Fourier%20Transform%20and%20Its%20Applications,%20Prentice%20Hall,%201988..md) [Schollwöck, U. Homework 1, Quantum Mechanics 1 (German) (2019-2020)](Schollwöck,%20U.%20Homework%201,%20Quantum%20Mechanics%201%20(German)%20(2019-2020).md) #MathematicalFoundations/Analysis/FourierAnalysis #MathematicalFoundations/Analysis/FourierAnalysis/Integrals #index #Bibliography