Mathematical Methods for Physics and Engineering - Matematica.NET

INTEGRAL TRANSFORMS13.4 Exercises13.1 Find the Fourier transform of the function f(t) =exp(−|t|).(a) By applying Fourier’s inversion theorem prove that∫π∞2 exp(−|t|) = cos ωt0 1+ω dω. 2(b) By making the substitution ω =tanθ, demonstrate the validity of Parseval’stheorem for this function.13.2 Use the general definition and properties of Fourier transforms to show thefollowing.(a) If f(x) is periodic with period a then ˜f(k) = 0, unless ka =2πn for integer n.(b) The Fourier transform of tf(t) isid˜f(ω)/dω.(c) The Fourier transform of f(mt + c) ise iωc/mm ˜f( ω).m13.3 Find the Fourier transform of H(x−a)e −bx ,whereH(x) is the Heaviside function.13.4 Prove that the Fourier transform of the function f(t) defined in the tf-plane bystraight-line segments joining (−T,0) to (0, 1) to (T,0), with f(t) = 0 outside|t|