# Why Are Sine and Cosine Called Harmonic Functions

I thought that the definition of a harmonic function f such that

$\nabla^2f=0$

In one dimension, doesn't this mean a function who's second derivative is zero? IE

$\frac{d^2f}{dy^2}=0$

However, the sine nor cosine function's second derivative does not equal zero, but in many textbooks they are refereed to as harmonic functions.

What's going on here?

• The might be referring to the complex plane. ​ ​ – user57159 Dec 9 '16 at 3:58
• in many textbooks they are refereed to as harmonic functions Is it the trig functions themselves which are called harmonic, or maybe some related context like harmonic motion? – dxiv Dec 9 '16 at 4:03
• What textbook specifically? In what context? – Robert Israel Dec 9 '16 at 4:55

Laplace's equation in polar coordinates takes the form $$\frac{\partial^2 f}{\partial r^2} + \frac{1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} = 0.$$ A first attempt at solving this is to assume that $f$ has the form $f(r,\theta) = u(r)v(\theta)$ for some functions $u$ and $v$; by adding together functions of this form, you hope to be able to deal with enough possible boundary conditions.
This procedure (separation of variables) leads to $$u''(r)v(\theta) + \frac{1}{r}u'(r) v(\theta) + \frac{1}{r^2}u(r)v''(\theta) = 0.$$ When $f$ is sufficiently nonzero, you can rearrange this to get $$\frac{r^2 u''(r) + ru'(r)}{u(r)} = -\frac{v''(\theta)}{v(\theta)} = \lambda,$$ a constant independent of $r$ and $\theta$; in particular, depending on the sign of $\lambda$, $v(\theta)$ will be some combination of $\sin$ and $\cos$.
$\sin$ and $\cos$ were already known as "harmonics" due to their appearance in the harmonic osciallator; the more general term "spherical harmonics" came later, and because of this connection all solutions of Laplace's equation eventually became known as "harmonic" functions.