Can I get the solution for these physics and mathmatics problems? textbook:

? Alec J. Schramm, "Mathematical Methods and Physical Insights"

Other helpful textbooks are:

? A. Altland, J. von Delft, "Mathematics for Physicists: Introductory Concepts and Methods"

? Mary L. Boas, "Mathematical Methods in the Physical Sciences"

Exercise 1. Fourier series [10 points] Consider the periodic function given by f (2) - 1 (1) a) Sketch this function. b) Compute the (trigonometric) Fourier series coefficients a,, and by, for f(r). Exercise 2. Fourier transform [15 points] Explicitly calculate the Fourier transform of the function f (t) = sin (wot) e-$2141, where wo and ? are positive real constants. The function f(t) is real. Check that the Fourier transform obeys the expected symmetry prop- erties. Hint: If you express the sine in the definition of f (t) through complex exponentials the integrals become simpler to evaluate. Exercise 3. Fourier transforms and differential equations [15 points] Consider the differential equation + as = 2at where a is a positive constant. a) Compute the corresponding Green's function in Fourier space, G(w), and its inverse Fourier transform G(t). b) Find a solution r(t) of the ODE using G(t) and the convolution theorem. Hint: You may use the following integral: du e but ate (t) 27 a - w where 9(1) - for t 2 0 for t - 0