Lecture 22 Appendix B (More on Legendre Polynomials and Associated Legendre functions)

Transcription

1 Lecture Appendix B (More on Legendre Polynomials and Associated Legendre functions) Ex.7: Let's use Mathematica to solve the equation and then verify orthogonality. First In[]:= Out[]= DSolve y'' x n^y x, y x, x y x C Cos nx C Sin nx y x C Cos nx C Sin nx As - expected the sines and cosines. Then consider the pair-wise integrals (the dot products) In[]:= Out[]= In[]:= Integrate Sin mx Sin nx, x,, ncos n Sin m mcos m Sin n m n ncos n Sin m mcos m Sin n m n FullSimplify, Assumptions m, n Integers Out[]= Interesting for m n, but how about m = n In[]:= Out[]= Integrate Sin mx Sin mx, x,, Sin m m Sin m m In[]:= Out[]= FullSimplify, Assumptions m Integers In[6]:= FullSimplify Integrate Cos mx Cos nx, x,,, Assumptions m, n Integers Out[6]= In[7]:= Out[7]= FullSimplify Integrate Cos mx Cos mx, x,,, Assumptions m Integers Finally

2 Lec App B.nb In[]:= Out[]= FullSimplify Integrate Sin mx Cos nx, x,,, Assumptions m, n Integers In[]:= FullSimplify Integrate Sin mx Cos mx, x,,, Assumptions m Integers Out[]= So always get zero here. Ex.7: Now consider the integrals with the Legendre polynomials In[]:= Integrate x^mlegendrep l, x, x,, Out[]= x m LegendreP l, x x Can't do the generic case. Make a table In[]:= TableForm Table m, l, Integrate x^mlegendrep l, x, x,,, m,,, l,, Out[]//TableForm= So, as expected, the integral is non-zero only when l m and both are either even or odd. The same structure for polynomials is

3 Lec App B.nb In[]:= TableForm Table m, l, Integrate LegendreP m, x LegendreP l, x, x,,, m,,, l,, Out[]//TableForm= So non-zero only for l = m. Ex.: Now expand a function with Legendre polynomials. Consider the function In[]:= In[]:= Out[]= F x : Sign x Plot F x, x,,, AxesLabel x, "F",. F..... x.. The expansion coefficients are given by

4 Lec App B.nb In[]:= In[6]:= c m : m Integrate LegendreP m, x F x, x,, Out[6]//TableForm= TableForm Table m, c m, m,, As expected, only the odd terms are non-zero. We can compare the values to our analytic results In[7]:= TableForm Table m, ^mgamma m m, m,, 7 Gamma m Gamma Out[7]//TableForm= Clearly in agreement. Next look at the truncated sums In[]:= In[]:= Out[]= FA N, x : Sum c m LegendreP m, x, m,, N FA, x x 7 6 x x

5 Lec App B.nb In[]:= Out[]= Not so good Plot, x,,, AxesLabel x, "F ", F x. In[]:= In[]:= Out[]= FA, x ; Plot, x,,, AxesLabel x, "F ", F x. Better In[]:= FA, x ;

6 6 Lec App B.nb In[]:= Out[]= Plot, x,,, AxesLabel x, "F ", F.... x OK, except for the issues at the endpoints. Ex.: 6 Consider the following function (singular at the origin) In[]:= In[6]:= Out[6]= F x : UnitStep x Log x ^ Plot F x, x,,, AxesLabel x, "F", LabelStyle Large, PlotStyle Thick, Exclusions None, PlotRange, F x Expand in Legendre polynomials In[7]:= c m : m Integrate LegendreP m, x F x, x,,

7 Lec App B.nb 7 In[]:= Out[]//TableForm= TableForm Table m, c m, m,, These results agree with the analytic ones - but it is clearly easier to use Mathematica here. In[]:= In[]:= In[]:= Out[]= FA N, x : Sum c m LegendreP m, x, m,, N FA, x ; Plot, x,,, AxesLabel x, "F ", F.... x Clearly need more terms, as expected due to the singular behavior. Ex.: Here we consider the polynomial In[]:= In[]:= F x : 7x^ x c m : m Integrate LegendreP m, x F x, x,,

8 Lec App B.nb In[]:= Out[]//TableForm= TableForm Table m, c m, m,, 6 7 All zero above as expected. In[]:= In[6]:= Out[6]= In[7]:= FA N, x : Sum c m LegendreP m, x, m,, N FA, x x x x x Simplify Out[7]= x 7x The desired polynomial. Ex.: Here we consider the polynomial In[]:= In[]:= In[]:= F x : x x^ c m : m Integrate LegendreP m, x F x, x,, TableForm Table m, c m, m,, Out[]//TableForm= 6 7 All zero above as expected, and only odd.

9 Lec App B.nb In[]:= FA N, x : Sum c m LegendreP m, x, m,, N In[]:= Out[]= In[]:= FA, x x x x Simplify Out[]= x x The desired polynomial. Ex.: Here we consider the function In[]:= F x : Abs x The coefficients in the expansion are In[]:= In[6]:= c m : m Integrate LegendreP m, x F x, x,, TableForm Table m, c m, m,, Out[6]//TableForm= Only the even terms are non-zero as expected. So the best fit quadratic polynomial is In[7]:= In[]:= Out[]= In[]:= Out[]= FA N, x : Sum c m LegendreP m, x, m,, N FA, x 6 x Simplify 6 x Make a comparison

10 Lec App B.nb In[]:= Out[]= Plot, Abs x, x,,, AxesLabel x, "F", F x Clearly a pretty good fit, but misses the details. Ex.: Now we want to consider the Associated Legendre function In[]:= LegendreP,, Cos Out[]= Cos Cos 7Cos By inspection this is what we obtained analytically except for the differing minus sign convention in Mathematica.