### 9.2.1 Defining periodic expressions

The periodic command creates periodic expressions.

- periodic takes two or four arguments:
  - expr, an expression f(x) where x∈[a,b⟩ for a<b.
  - x, a real variable.
  - a, a real number specifying the lower bound for x.
  - b, a real number specifying the upper bound for x.
  The last three arguments can be passed as a single argument x=a..b.
- periodic(expr,x,a,b) or periodic(expr,x=a..b) returns a periodic expression

  g(x)=f | ⎛ ⎜ ⎜ ⎝ | x−(b−a) | ⎢ ⎢ ⎢ ⎣ |
  x−a
  b−a
  | ⎥ ⎥ ⎥ ⎦ | ⎞ ⎟ ⎟ ⎠ | , x∈ℝ,

  with period T=b−a, satisfying the property g(x+T)=g(x) for all x∈ℝ.

#### Example

To define the periodic function f(x)=1−x2 for −1≤ x<1 with period T=2, input:

f:=periodic(1-x^2,x,-1,1)

or:

f:=periodic(1-x^2,x=-1..1)

− | ⎛ ⎜ ⎜ ⎝ | −2 | ⎢ ⎢ ⎢ ⎣ |
x+1
2
| ⎥ ⎥ ⎥ ⎦ | +x | ⎞ ⎟ ⎟ ⎠ |
2
| +1

Indeed:

plot(f,x=-5..5)