# 4. Half Range Fourier Series

If a function is defined over half the range, say 0 to L, instead of the full range from -L to L, it may be expanded in a series of sine terms only or of cosine terms only. The series produced is then called a half range Fourier series.

Conversely, the Fourier Series of an even or odd function can be analysed using the half range definition.

## Even Function and Half Range Cosine Series

An even function can be expanded using half its range from

• 0 to L or
• -L to 0 or
• L to 2L

That is, the range of integration is L. The Fourier series of the half range even function is given by:

f(t)=a_0/2+sum_(n=1)^oo\ a_n\ cos (n pi t)/L

for n = 1, 2, 3, ... , where

a_0=2/Lint_0^Lf(t)dt

a_n=2/Lint_0^Lf(t)\ cos (n pi t)/Ldt

and b_n= 0.

### Illustration

In the figure below, f(t) = t is sketched from t = 0 to t = π.

An even function means that it must be symmetrical about the f(t) axis and this is shown in the following figure by the broken line between t = -π and t = 0.

It is then assumed that the "triangular wave form" produced is periodic with period 2pi outside of this range as shown by the dotted lines.

### Example

We are given that

f(t)={: {(-t,if, -pi<=t<0),(t,if, 0<=t < pi):}

and f(t) is periodic with period 2π.

a) Sketch the function for 3 cycles.

b) Find the Fourier trigonometric series for f(t), using half-range series.

## Odd Function and Half Range Sine Series

An odd function can be expanded using half its range from 0 to L, i.e. the range of integration has value L. The Fourier series of the odd function is:

Since ao = 0 and an = 0, we have:

f(t)=sum_(n=1)^oo\ b_n\ sin (n pi t)/L for n = 1, 2, 3, ...

b_n=2/Lint_0^Lf(t)\ sin (n pi t)/Ldt

In the figure below, f(t) = t is sketched from t = 0 to t = π, as before.

An odd function means that it is symmetrical about the origin and this is shown by the red broken lines between t = −π and t = 0.

It is then assumed that the waveform produced is periodic of period 2π outside of this range as shown by the dotted lines.

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