Question

(a) What kind of functions can be decomposed into Fourier series and what 10 into Fourier integral? man (b) When analysing stability of equilibrium states in various physical prob- lems, why do we suppose variations about equilibrium values to be Small? (c) In which physical problems do we come across Bessel's equations (men- tion one or two problems commenting on symmetry properties of solu- tions)? (d) When solving the wave or heat equation by separation of variables, we first seek particular solutions in the form F(x)G(t) and then sum them up. Why are we sure that the sum satisfies the equation? (Comment on a property of the governing equation).

Answer 1

Anscoer Pried fonctions Fourier provedt that signal cuth periool T ole compoced into sinusoi dlal 'Rnetions be decomposeol toto Foorier Can Campasite pericolie be Series CAngueney f Can Neo-peciodie foretions Forier integral. Frenges Continuous set of valuec. be decompaseal into a Over DWhile analysing Phiy sical problems, Variations about eguilibrium stability of equilikium states io Values small, lenaling Smaller the listance of mo tion the equilibrium point, esuilibrium So that the change x is Lo small deviation of the motion oscillabion Ore 11ows near 7inest easily Bessel fnction 3-limensional fugqerncy Useol to salve in are equation at a given electromagnetem wave It is use.d in

al eguation,cohich arises y lindrical coor eli nates Bessel's differenti e specially in polar anol defneed It be QS con 2 + ir& Bessels fonetion of ordder n 2 when n is atve real numbers.