What are Parseval's Identities?
Channel: MechanicaLEi
Date: 2018-12-02

Welcome back MechanicaLEi, did you know that Parseval's Identity was named after Marc-Antoine Parseval and is known as the Pythagorean theorem for inner-product spaces? This makes us wonder, what are Parsevel's Identities?
Before we jump in check out the previous part of this series to learn about what half sine and cosine series are? 
For all f of x equal to summation of b n into sin of n pi x upon L from n equals to 1 to infinity, where x lies in the open interval of zero comma L; Parseval's Identity states that 2 upon L into integral of f of x squared dx from zero to L is equal to summation of b squared n from n equals to one to infinity. For all undergraduate purposes this identity is accepted without proof. 
Let a function f of x be defined on the closed interval of minus pi comma pi. Using the well known Euler's formulas, cos phi equals to e power i phi plus e power minus i phi upon two and sin phi equals to e power i phi minus e power minus i phi upon 2; we can write the Fourier series of the function in complex form as f of x equals to summation of c n into e power i n x from n equals minus infinity to infinity. c n is called complex Fourier coefficient and is defined by the formula c n equals to one upon 2 pi into integral of f of x into e power minus i n x dx from minus pi to pi. Here, n may take any integer value. However, If necessary, to expand a function f of x of period 2L we can simply use the following expressions by simply multiplying the f of x equation's power with pi upon L and replacing the intervals of pi with L. 
Hence, we first saw what Parseval's Identities are and then went on to see what Complex form of Fourier series are?

In the next episode of MechanicaLEi find out what Partial differential equations are?


Attributions:

Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/)
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100255
Artist: http://incompetech.com/

Subtle Library by Fabian Measures (http://freemusicarchive.org/music/Fabian_Measures/) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/by/4.0/)
Source: http://freemusicarchive.org/music/Fabian_Measures/Emotion_Music/Subtle_Library