Power Series and Transcendental Functions

Imagine that you are an 18th-Century astronomer, hard at work on Celestial Mechanics. Understanding and predicting planetary motions requires calculating huge numbers of trigonometric functions such as sine and cosine. Obviously, this is before the age of computers or calculators, or mathematical tables.

How would you compute the sine, the cosine, or the tangent ..., of some angle???

(How did they do it??)

Functions such as sine, cosine, logarithm, exponential, hyperbolic cotangent, and so forth, are called transcendental functions. They are defined in terms of the limits of power series: infinite series of terms involving the argument of the function (the argument of f(x) is x; the argument of $\cos(\theta)$ is $\theta$), raised to an integer power, with associated coefficients in front. Here are some examples of power series that define transcendental functions:

$$\exp(\theta) 1 + \frac{\theta}{1!} + \frac{\theta^{2}}{2!} + \frac{\theta^{3}}{3!} + \cdots + \frac{\theta^{n}}{n!} + \cdots ,$$ = (1)

$$\log(1+\theta) \theta - \frac{\theta^{2}}{2} + \frac{\theta^{3}}{3} - \frac{\theta^{4}}{4} + \frac{\theta^{5}}{5} - \cdots ,$$ = (2)

$$\tan(\theta) \theta + \frac{\theta^{3}}{3} + 2\frac{\theta^{5}}{15} + 17\frac{\theta^{7}}{315} + 62\frac{\theta^{9}}{2835} + \cdots ,$$ = (3)

$$\cos(\theta) 1 - \frac{\theta^{2}}{2!} + \frac{\theta^{4}}{4!} - \frac{\theta^{6}}{6!} + \cdots ,$$ = (4)

$$\sin(\theta) \theta - \frac{\theta^{3}}{3!} + \frac{\theta^{5}}{5!} - \frac{\theta^{7}}{7!} + \cdots ,$$ = (5)

$$\coth(\theta) \frac{1}{\theta} + \frac{\theta}{3} - \frac{\theta^{3}}{45} + 2\frac{\theta^{5}}{945} - \frac{\theta^{7}}{4725} + \cdots ,$$ = (6)

Such expressions - truncated after a certain number of terms - are precisely how computers and calculators evaluate these functions. There is no other way to do it! That is why, if you were the Principal Assistant to the Astronomer Royal in 1720, you spent all of your time with ink quill and paper calculating endless power series such as the above.... :-(