{{page>wiki:headers:hheader}}

%% Instructions:
%% 1. Replace "The Question" with the question.
%% 2. Change mmswzzz everywhere (2x) to the basename of this file.
%% 3. Upload the PDF and PPT versions of the question.
%%    (Make sure to use the same basename.)
%% 4. Change xx in the 2-letter acronym in the course footer to reference
%%    the correct course.
%% 5. Delete these instructions when done!


====== Calculations with Legendre's Function ======

===== The Prompt =====

**How would Legendre's function (shown below) change with the coordinate transformation $z = cos(\theta)$**

$$
\left[\sin{\theta}\frac{d}{d\theta}\left(\sin{\theta}\frac{d}{d\theta}\right)-A\sin{\theta}^{2}\right]\Theta(\theta)=0
$$
 {{swbq:cfsw:cfswleg1.ppt|Powerpoint slide}}
\\\\
{{swbq:cfsw:cfswleg1.pdf|PDF slide}}


**How is** $\frac{d}{d\theta}$ **related to** $\frac{d}{dz}$ **by our previous coordinate transformation?**

{{swbq:cfsw:cfswleg2.ppt|Powerpoint slide}}
\\\\
{{swbq:cfsw:cfswleg2.pdf|PDF slide}}


**Do the product rule and write things in terms of first and second derivatives of z with our newly transformed Legendre function. **
$$
\left[\left(1-z^{2}\right)\frac{d}{dz}\left(1-z^{2}\right)\frac{d}{dz}-A\left(1-z^{2}\right)\right]\Theta(z)=0
$$
{{swbq:cfsw:cfswleg3.ppt|Powerpoint slide}}
\\\\
{{swbq:cfsw:cfswleg3.pdf|PDF slide}}


**How do we find the coefficients of the expression? **
$$
f(z)=\sum_{l=0}^{\infty}c_{l}P_{l}(z)
$$

{{swbq:cfsw:cfswleg4.ppt|Powerpoint slide}}
\\\\
{{swbq:cfsw:cfswleg4.pdf|PDF slide}}

===== Context =====

This [[strategy:smallwhiteboard:|SWBQ]]

===== Wrap Up =====

FIXME - Fix organizational structure here.

{{page>wiki:footers:courses:cffooter}}

{{page>wiki:footers:topics:mmfooter}}