Question

fourier analysis

2. (a) Find the Fourier sine transform of b) Write f(x) as an inverse sine transform Hint: Don't directly calculate F,[f (x)(w). Begin with showing the representation sin wxdw. x >0 ㄧㄨ and then interchanging x and w in the representation. Now look at it carefully, what does the equation tell you?

Answer 1

By definition F_e {1/1 + x^2} = integral_0^infinity cos sx dx/1 + x^2 dx = I, (say) ....(1) therefore dI/ds = integral_0^infinity - x sin sx dx/1 + x^2 = -integral_0^infinity x sin sx dx/1 + x^2 ...(2) or, dI/ds = -integral_0^infinity x^2 sin sx dx/x(1 + x^2) = -integral_0^infinity [(1 + x^2) - 1]sin sx dx/x(1 + x^2) = -integral_0^infinity sin sx dx/x + integral_0^infinity sin sx dx/x(1 + x^2) therefore dI/ds = -pi/2 + integral_0^infinity sin sx dx/x(1 + x^2) as integral_0^infinity sin sx dx/x = pi/2 ...(3) from 3, d^2 I/ds^2 = integral_0^infinity x cos sx dx/x(1 + x^2) = integral_0^infinity cos sx/1 + x^2 = I. or, d^2I/ds^2 - I = 0, or, (D^2 - 1)I = 0, where D = d/ds chose solution in I = Ae^-s + Be^s ...(4) (A, B being arbitrary constant) \r\n from 4, dI/ds = -Ae^- s + Be^s ...(5)