Reduction formulas by computer

You will turn three reduction formulas into code. Each one builds a value from a smaller case, with a loop that starts from the easy case.

The idea. A reduction formula gives $I_n$ from $I_{n-1}$ or $I_{n-2}$. Start with the easy cases and apply the formula one step at a time.

• Wallis: $W_n = \dfrac{n - 1}{n}W_{n-2}$ with $W_0 = \dfrac{\pi}{2}$ and $W_1 = 1$.
• Powers times $e^x$: $I_n(x) = x^n e^x - n\,I_{n-1}(x)$ with $I_0(x) = e^x$.
• Tangent: $T_n = \dfrac{1}{n - 1} - T_{n-2}$ with $T_0 = \dfrac{\pi}{4}$ and $T_1 = \dfrac{1}{2}\ln 2$.

Your tasks.

1. Finish wallis(n). Return $W_n$.
2. Finish xexp(n, x). Return $I_n(x)$, with the constant taken as $0$.
3. Finish tangent_integral(n). Return $T_n$.