nLab cotangent function

Idea

In trigonometry the cotangent function is one of the basic trigonometric functions.

Definition

The contangent is the ratio of the cosine and the sine:

$$\cot(x) = \frac{\cos(x)}{\sin(x)}$$

Properties

Relations with tangent function

• The cotangent and tangent are reciprocal to each other: $\cot x = \frac1{\tan x}$.

• The cotangent and tangent are complementary to each other: $\cot x = \tan (\frac{\pi}{2} - x)$.

• Double angle formula: $\tan x = \cot x - 2\cot 2x$.

Relation to Bernoulli numbers

The hyperbolic analog $\coth x = \frac{e^x + e^{-x}}{e^x - e^{-x}}$ is related to $\cot x$ via the formula

$$\cot x = i\coth i x.$$

Meanwhile $\coth x$ is related to the Bernoulli numbers $B_n$, defined by the exponential generating function

$$\frac{x}{e^x - 1} = \sum_{n \geq 0} \frac{B_n x^n}{n!},$$

through a series of equations

$$\frac{x}{e^x - 1} + \frac{x}{2} = \frac{x(2 + e^x - 1)}{2(e^x - 1)} = \frac{x}{2} \frac{e^x + 1}{e^x - 1} = \frac{x}{2} \frac{e^{x/2} + e^{-x/2}}{e^{x/2} - e^{-x/2}} = \frac{x}{2}\coth \frac{x}{2}.$$

Notice the right side defines an even function. Therefore

$$\frac{x}{2}\coth \frac{x}{2} = \sum_{n \geq 0} \frac{B_{2n} x^{2n}}{(2n)!}$$

and so

$$x \cot x = i x \coth i x = \sum_{n \geq 0} \frac{B_{2n} (2i x)^{2n}}{(2n)!} = \sum_{n \geq 0} (-1)^n \frac{B_{2n} 2^{2n} x^{2n}}{(2n)!}.$$

“Eisenstein” series expansion

The cotangent is the logarithmic derivative? of the sine function:

$$\cot x = (\log (\sin x))'.$$

Applying this observation to the Euler-Weierstrass product formula for the sine function (see there for a proof):

$$\sin (\pi x) = \pi x \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)$$

one obtains the following summation formula for the cotangent:

$$\pi\, \cot (\pi x) = \frac1{x} + \sum_{n=1}^\infty \left(\frac1{x + n} + \frac1{x - n}\right)$$

This expansion was used by Eisenstein as a starting point for developing the theory of trigonometric functions; Eisenstein’s account of elliptic functions (cf. the eponymous Eisenstein series), developed further by Weierstrass, Kronecker, and others, runs parallel to his trigonometric theory, as explained later by Weil. For some more details, see these notes by Varadarajan.

Relation to zeta function values

Proposition

The power series identity

$$\pi x \cot \pi x = 1 - 2\sum_{k \geq 1} \zeta(2k) x^{2k}$$

holds over an open domain where the series converges, ${|x|} \lt 1$.

Proof

From the Eisenstein expansion, we have

$$\pi x\, \cot (\pi x) = x\, \left(\frac1{x} + \sum_{n=1}^\infty \left(\frac1{x + n} + \frac1{x - n}\right)\right) = 1 + \sum_{n \geq 0} \frac{2x^2}{x^2 - n^2} = 1 - 2\sum_{n \geq 1} \frac{x^2/n^2}{1 - x^2/n^2}.$$

By a geometric series expansion, the last expression is

$$1 - 2\sum_{n \geq 1} \sum_{k \geq 1} \left(\frac{x^2}{n^2}\right)^k = 1 - 2\sum_{k \geq 1} x^{2k} \sum_{n \geq 1} \frac1{n^{2k}}$$

which is the same as $1 - 2\sum_{k \geq 1} \zeta(2k)x^{2k}$.

Related concepts

• tangent function

• trigonometric identity

• Eisenstein series

• product formula for the sine function

References

• Wikipedia, Trigonometric functions – tan

• Veeravalli Varadarajan, Circular Functions (pdf).