Power series of tan(x), cot(x), csc(x)

Here’s a little how-to on figuring out the power series of tan(x), cot(x) and csc(x).

Start with the generating function for the Bernoulli numbers:

$\displaystyle \sum_{k=0}^{\infty} B_k \frac{t^k}{k!} = \frac{t}{e^t-1}$

Take only the even powers of t on the LHS:

$\displaystyle \sum_{k=0}^{\infty} B_k \frac{t^k}{k!} + \sum_{k=0}^{\infty} B_k \frac{(-t)^k}{k!} = \frac{t}{e^t-1} - \frac{t}{e^{-t}-1}$

$\displaystyle 2 \sum_{k=0}^{\infty} B_{2k} \frac{t^{2k}}{(2k)!} = \frac{t\left(e^{-t}-e^t\right)}{\left(e^t-1\right)\left(e^{-t}-1\right)}$

$\displaystyle = \frac{t\left(e^{-\frac{1}{2} t}+e^{\frac{1}{2} t}\right)\left(e^{-\frac{1}{2} t}-e^{\frac{1}{2} t}\right)}{e^{\frac{1}{2} t}\left(e^{\frac{1}{2} t}-e^{-\frac{1}{2} t}\right)e^{-\frac{1}{2} t}\left(e^{-\frac{1}{2} t}-e^{\frac{1}{2} t}\right)}$

$\displaystyle = \frac{t\left(e^{\frac{1}{2} t}+e^{-\frac{1}{2} t}\right)}{e^{\frac{1}{2} t}-e^{-\frac{1}{2} t}}$

Put $t=2ix$ :

$\displaystyle 2 \sum_{k=0}^{\infty} B_{2k} \frac{(2ix)^{2k}}{(2k)!} = \frac{2ix\left(e^{ix}+e^{-ix}\right)}{e^{ix}-e^{-ix}}$

$\displaystyle \sum_{k=0}^{\infty} B_{2k} \frac{(-1)^k 4^k x^{2k-1}}{(2k)!} = \frac{i\left(e^{ix}+e^{-ix}\right)}{e^{ix}-e^{-ix}}$

Recall that $\sin x = \frac{e^{ix}-e^{-ix}}{2i}$ and $\cos x = \frac{e^{ix}+e^{-ix}}{2}$ :

$\displaystyle \frac{\cos x}{\sin x} = \cot x = \sum_{k=0}^{\infty} B_{2k} \frac{(-1)^k 4^k x^{2k-1}}{(2k)!}$

That was pretty easy. How about tan and csc?

Consider

$\displaystyle \cot x - 2\cot 2x = \cot x - \frac{\cot^2 x - 1}{\cot x}$

$\displaystyle = \frac{1}{\cot x}$

$\displaystyle = \tan x$

$\displaystyle \tan x = \sum_{k=0}^{\infty} B_{2k} \frac{(-1)^k 4^k x^{2k-1}}{(2k)!} - 2\sum_{k=0}^{\infty} B_{2k} \frac{(-1)^k 4^k 2^{2k-1} x^{2k-1}}{(2k)!}$

$\displaystyle \tan x = \sum_{k=0}^{\infty} B_{2k} \frac{(-1)^k 4^k (1-4^k) x^{2k-1}}{(2k)!}$

And finally,

$\displaystyle \cot x - \cot 2x = \cot x - \frac{\cot^2 x - 1}{2\cot x}$

$\displaystyle = \frac{\csc^2 x}{2\cot x}$

$\displaystyle = \frac{1}{2 \sin x \cos x}$

$\displaystyle = \csc 2x$

$\displaystyle \csc x = \sum_{k=0}^{\infty} B_{2k} \frac{(-1)^k 4^k x^{2k-1}}{2^{2k-1} (2k)!} - \sum_{k=0}^{\infty} B_{2k} \frac{(-1)^k 4^k x^{2k-1}}{(2k)!}$

$\displaystyle \csc x = \sum_{k=0}^{\infty} B_{2k} \frac{(-1)^k (2-4^k) x^{2k-1}}{(2k)!}$

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This entry was posted on Sunday, October 30th, 2011 at 10:57 am and is filed under mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Responses to Power series of tan(x), cot(x), csc(x)

fyyre says:

November 2, 2011 at 2:51 am

More code, less math.

Reply
drac says:

November 7, 2011 at 11:53 am

Thanks, Nice little piece =)

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