The solution is a one page document written in Word and using Mathtype for the equations describing (and proving) how the Fourier coefficients of a periodic function are connected to the Fourier coefficients of the derivative of the function. This method uses purely real integrals as opposed to the alternative method using complex integration.

... theory are exactly the same as for Fourier theory: in ... we use it to find the coefficients needed in our ... itself only uses first and second derivatives, so one ...

... 0, since upper limit of the integral is zero) By applying Leibniz formula (derivative of an ... Thus, the coefficients Dn are given by Fourier formulas: T π ...

... original equation, the partial derivatives become regular derivatives: ... to determine what are the constant coefficients . ... is known as the Fourier sine expansion ...

... be expressed by means of Fourier law: ( 2 ... constant () which is called sometimes "thermal diffusivity coefficient". ... m = -1, the first derivative of function (v ...

... so the partial derivatives become full derivatives and the ... of the eigenfunctions , which according to Fourier sine series ... we must have the coefficients of the ...

... By Fourier's theorem, any complex wave can be considered ... with an inversion (or a reflection coefficient of -1 ... where and are derivatives of the two function with ...

... The general Fourier expansion of the function f(x) on ... an even n are zero, while the coefficients with an ... The derivative of the absolute value is not defined at ...

... Taking the derivative with respect to n: ... However, we can expand f(x) as a Fourier series ... Since this is a power series, all the coefficients of the nth term must ...

... The Fourier transform of the nth derivative is: ... harmonic oscillator (remember that now k now is constant in the equation, therefore the coefficients may be ...

... as the overshoot in the truncated Fourier series representation ... we have to worry only about the coefficients of the ... And the leading coefficient is: > a[0]:=1/Pi ...