Even and Odd Functions

You probably can recall the definitions of even and odd functions:

A function is even if f(-x)=f(x) for all x.

Note that this implies that the graph of f(x) is symmetric with respect to the vertical axis.

A function is odd if f(-x)=-f(x) for all x.

Note that this implies that the graph of f(x) is symmetric with respect to the origin.

Examples of even functions are: constant functions, x^k with k an even integer, cos(x).
Examples of odd functions are: x^k with k an odd integer, sin(x).

In[24]:=

  Plot[Evaluate[Table[x^k,{k,0,10,2}]],{x,-3,3},
  PlotRange->{-.5,10}, PlotLabel->
  FontForm["x^k, k even",{"Helvetica-Bold",12}]];

In[25]:=

  Plot[Evaluate[Table[x^k,{k,1,10,2}]],{x,-3,3},
  PlotRange->{-10,10}, PlotLabel->
  FontForm["x^k, k odd",{"Helvetica-Bold",12}]];

You probably can also recall from calculus that if f(x) has a Taylor series expansion about x0=0 containing only even/odd powers, then f(x) is an even/odd function.

In[26]:=

  Series[Sin[x],{x,0,10}]

Out[26]=

       3    5      7       9
      x    x      x       x          11
  x - -- + --- - ---- + ------ + O[x]
      6    120   5040   362880

In[27]:=

  Series[Cos[x],{x,0,10}]

Out[27]=

       2    4    6      8        10
      x    x    x      x        x           11
  1 - -- + -- - --- + ----- - ------- + O[x]
      2    24   720   40320   3628800

By using the definitions of even and odd functions, you should convince yourself of the following:
a. The sum of even (odd) functions is even (odd).
b. The product of two even or two odd functions is even.
c. The product of an even and an odd function is odd.

I'll prove the last one for you.
Let f be even and g be odd.
fg(-x)=f(-x)g(-x)=f(x)[-g(x)]=-f(x)g(x)=-fg(x)

You should also convince yourself (or recall from calculus) that the integral from -T to T of an odd function is zero and the integral from -T to T of an even function is twice the integral from 0 to T.

These two facts can greatly simplify the calculation of Fourier coeficients.

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