Question

Green's function

2 The Green function (10 P) The Fourier transform plays a tantamount role in the theory of inhomogeneous, linear differential equations. If as was shown in the lecture - G is a so called fundamental solution of the differential equation CG(z,z') = δ(z-z') one may calculate a particular solution for an inhomogeneity g by convolution G is called Green function. Since the Fourier transform maps derivatives to multiplications, it simplifies the calcu- lation of the Green function to an algebraic problem and subsequent Fourier inversion. An example a) (2P) Again, we look at the damped harmonic oscillator. Let We are interested in the behavior of the oscillator under the influence of an arbi trary driving force1 g(t) Map (1) by Fourier transformation to an algebraic equation and determine the Green function G(t, t) up to execution of the inversion integral. We are going to explicitly solve this integral later with methods from complex calculus

Answer 1

w (t) f we getutin ん&CT) od 1 wT od ˊ町 -시

WT 2不 4 dT GCT) -여 UUT dT 예 Poles one dejined ba

2 LJT GT)- wT 2.

Residve uheste 2巳 14ヶ CA AT The ntegal et R-→ 서

Svniar CA in