Fixing Normalization Issues in Graphing Orthonormalized Functions

[scigal89]

In class we worked out the following

<br /> <br /> \int e^{ik(x-X)}dk=\frac{e^{ik(x-X)}}{i(x-X)}\approx \frac{sin[k(x-X)]}{x-X}<br /> <br />

by taking the real part of the solution. My teacher wants us to graph the following functions

<br /> <br /> \psi_{1} \sim \frac{sin(x)}{x}<br /> <br />

<br /> <br /> \psi_{2} \sim \frac{sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{2x}<br /> <br />

The second function, though, has a little dip in it at 0 that shouldn't be there. He says that's due to the fact that the functions aren't normalized and that we should be doing Gram-Scmidt or some other procedure to obtain the proper results. However, when I do Gram-Schmidt I get something nasty that can't possibly be correct (and most importantly, doesn't correct the problem). So what am I doing wrong?

 

[scigal89]

For Gram-Schmidt I take interval as (0, 2π). Then

<br /> <br /> \phi _{0} = \frac{sin(x)}{x\sqrt {Si(4 \pi)}}<br /> <br />

<br /> <br /> \psi _{1} = \left \{ \frac{sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{(2x)}\right \}-\left [ \int_{0}^{2\pi}\left \{ \frac{sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{(2x)}\right \}\frac{sin(x)}{x\sqrt {Si(4 \pi)}}dx \right ]\frac{sin(x)}{x\sqrt {Si(4 \pi)}}<br /> <br />

<br /> <br /> \psi _{1} \sim \frac{0.2627sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{2x}<br /> <br />

<br /> <br /> \phi _{1} \sim 3.3631\left \{ \frac{0.2627sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{2x} \right \}<br /> <br />

using the procedure for Gram-Schmidt outlined here: http://mathworld.wolfram.com/Gram-SchmidtOrthonormalization.html

and with a weighting function of 1.

 

[scigal89]

Any suggestions?