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Odd and Even Functions

The trigonometric functions have the additional properties of being an odd function about the point $x=0$ : $f_{\text{odd}}: f_{\text{odd}}(x) = -f_{\text{odd}}(-x)$ in the case of the sine, and an even function in the case of the cosine: $f_{\text{even}}: f_{\text{even}}(x) = f_{\text{even}}(-x)$ .

This can generalized to say that a function is even or odd about a point $\lambda/2$ : $f_{\text{odd} \frac{\lambda}{2}}: f_{\text{odd} \frac{\lambda}{2}}(\lambda/2 + x) = -f_{\text{odd} \frac{\lambda}{2}}(\lambda/2 -x)$ and $f_{\text{even} \frac{\lambda}{2}}: f_{\text{even} \frac{\lambda}{2}}(\lambda/2 + x) = f_{\text{even} \frac{\lambda}{2}}(\lambda/2 -x)$ .

Any function can be decomposed into an odd and even sum:

$$g(x) = g_{\text{even}} + g_{\text{odd}}$$ (17-3)

The sine and cosine functions can be considered the odd and even parts of the generalized trigonometric function:

$$e^{i x} = \cos(x) + \imath \sin(x)$$ (17-4)

with period $2 \pi$ .