 Welcome back mechanically AI. Did you know that Parsville's identity was named after Mark Antoine Parsville and is known as the Pythagorean theorem for inner product spaces? This makes us wonder, what are Parsville'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 bn into sine of n pi x upon l from n equals 1 to infinity where x lies in the open interval of 0 comma l, Parsville's identity states that 2 upon l into integral of f of x squared dx from 0 to l is equal to summation of b squared n from n equals 1 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 2 and sine 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 cn into e power i n x from n equals minus infinity to infinity. cn is called the complex Fourier coefficient and is defined by the formula cn equals to 1 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. When necessary to expand a function f of x of period 2 l, we can reuse the previous expression by simply multiplying the f of x equations power with pi upon l and replacing the intervals of pi with l. Hence we first saw what Parsville's identities are and then went on to see what complex form of Fourier series are. So like, subscribe and comment with your feedback to help us make better videos. Thanks for watching. Also thanks a lot for those constructive comments. You helped the channel grow. So here are the top mechanical EIs of our last videos. In the next episode of Mechanical EI, find out what partial differential equations are.