Mathematical Tools for Physics - Department of Physics - University

4—Differential Equations 94 each case explain why the result is as it should be. Ans:(F 0 /m)[−cosω 0 t+ cosωt]/(ω^20 −ω^2 ) 4.9 ...

4—Differential Equations 95 4.20 Solve by Frobenius series solution aboutx= 0: y′′+xy= 0. Ans: 1 −(x^3 /3!) + (1. 4 x^6 /6!)−(1. ...

4—Differential Equations 96 4.30 Verify that the equations (4.52) really do satisfy the original differential equations. 4.31 Wh ...

4—Differential Equations 97 4.37 For a second order differential equation you can pick the position and the velocity any way tha ...

4—Differential Equations 98 4.45 In the equation of problem4.17, make the change of independent variablex= 1/z. Without actually ...

4—Differential Equations 99 4.56 What is the total impedance left to right in the circuit R 1 R 2 C 1 L 1 C 2 L 2 ? Ans:R 1 + (1 ...

Fourier Series . Fourier series started life as a method to solve problems about the flow of heat through ordinary materials. It ...

5—Fourier Series 101 The second form doesn’t work as well as the first one, and there’s a reason for that. The sine functions al ...

5—Fourier Series 102 There are orthogonality relations similar to the ones forxˆ,yˆ, andˆz, but for sines and cosines. Letnandmr ...

5—Fourier Series 103 Interchange the order of the sum and the integral, and the integral that shows up is the orthogonality inte ...

5—Fourier Series 104 (What happens to the series Eq. (5.7) if you multiply everyunby 2? Nothing, because the coefficients anget ...

5—Fourier Series 105 −L L Here the discontinuity in the sine series is more obvious, a fact related to its slower convergence. 5 ...

5—Fourier Series 106 Now do two partial integrations. Work on the second term on the left: ∫b a dxu 1 u 2 ′′=u 1 u 2 ′ ∣ ∣ ∣ b a ...

5—Fourier Series 107 Apply the Theorem As an example, carry out a full analysis of the case for whicha= 0andb=L, and for the bou ...

5—Fourier Series 108 Notice that in this case the indexnruns over all positive and negative numbers and zero, not just the posit ...

5—Fourier Series 109 5.4 Musical Notes Different musical instruments sound different even when playing the same note. You won’t ...

5—Fourier Series 110 Real musical sound is of course more than just these Fourier series. At the least, the Fourier coefficients ...

5—Fourier Series 111 A bonus. You have the sum of this infinite series, a result that would be quite perplexing if you see it wi ...

5—Fourier Series 112 Substitute the assumed form and it will determineA. [ m(−ω^2 e) +b(iωe) +k ] Aeiωet=eiωet This tells you th ...

5—Fourier Series 113 All there is to do now is to solve for an inhomogeneous solution one term at a time and then to add the res ...