# Calculations with Legendre's Function

## The Prompt

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

$$\left[\sin{\theta}\frac{d}{d\theta}\left(\sin{\theta}\frac{d}{d\theta}\right)-A\sin{\theta}^{2}\right]\Theta(\theta)=0$$ Powerpoint slide
PDF slide

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

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

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

This SWBQ

## Wrap Up

- Fix organizational structure here.

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