 Welcome back MechanicalEI, did you know that half-range sine and cosine series are so-called because their expansion literally halves the expansion range? This makes us wonder, what are half-sine and cosine series? Before we jump in, check out the previous part of this series to learn about what Dirichlet's conditions are. First, let's clear out a few prerequisites. A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f of minus x equals to f of x for all x in the domain of f. It is also represented as f of x minus f of minus x equals to 0. On the other hand, a function f is odd if the graph of x is symmetric with respect to the origin. Algebraically, f is odd if and only if f of minus x is equal to minus f of x for all x in the domain of f. It is also represented as f of x plus f of minus x is equal to 0. Now, if f of x is an even function on open interval of minus p comma p, f of x can be expanded to an even periodic function with period 2p. The Fourier series expansion of such a function is given by f1 of x equals to a0 upon 2 plus summation of an into cos of n pi x upon p from n equals 1 to infinity, where a0 equals to 2 upon p into integral of f of x dx from 0 to p and an equals to 2 upon p into integral of f of x into cos of n pi x upon dx from 0 to p. This is called the half range cosine series of f of x. Similarly, for an odd function with the same open interval of minus p to p, the Fourier series f2 of x is given by summation of bn into sin of n pi x upon p from n equals 1 to infinity. Here, bn is given by 2 upon p into integral of f of x into sin of n pi x upon p dx from 0 to p. This expansion is called the half sine series expansion. Note that the domain has been halved from open interval of minus p comma p to the range of 0 to p, actually giving the name half range to the expansion series. Hence, we first saw what even and odd functions are and then went on to see what half sine and cosine series are. 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 Parswell's identities are.