6.16.3  Defining algebraic functions

Defining a function from ℝ^p to ℝ^q

If expr is an expression possibly involving a variable x, you can use it to define a function f either by
f(x):=expr
or f := x->expr
(see Section 5.5.1).

Warning!!!
The expression after -> is not evaluated. You should use unapply (see Section 6.15.2) if you expect the second member to be evaluated before the function is defined.

Example.
To define f:(x)→ x*sin(x),
Input:

or:

then:

Output:

You can similarly define a function of several variables, by replacing x by a sequence (x₁,…,x_p) or a list [x₁,…,x_p] of variables.

Example.
Input:

or:

then:

Output:

You can also define a function with values in ℝ^q by replacing expr by a sequence (expr₁,…,expr_q) or list [expr₁,…,expr_q] of expressions.

Examples.

• Define the function h (x,y)→ (x*cos(y),x*sin(y)).
  Input:
  h(x,y):=(x*cos(y),x*sin(y))
  then:
  h(2,pi/4)
  Output:

  √

  2

  ,

  √

  2

• Define the function h (x,y)→ [x*cos(y),x*sin(y)].
  Input:
  h(x,y):=[x*cos(y),x*sin(y)];
  or:
  h:=(x,y)->[x*cos(y),x*sin(y)];
  or:
  h(x,y):={[x*cos(y),x*sin(y)]};
  or:
  h:=(x,y)->return[x*cos(y),x*sin(y)];
  or:
  h(x,y):={return [x*cos(y),x*sin(y)];}
  then:
  h(2,pi/4)
  Output:

  ⎡ ⎢ ⎣

  √

  2

  ,

  √

  2

  ⎤ ⎥ ⎦

Defining families of function from ℝ^p−1 to ℝ^q using a function from ℝ^p to ℝ^q

Suppose that the function f: (x,y) → f(x,y) is defined, and you want to define a family of functions g(t) such that g(t)(y):=f(t,y) (i.e. t is viewed as a parameter). Since the expression after -> (or :=) is not evaluated, you should not define g(t) by g(t):=y->f(t,y); you have to use the unapply command (see Section 6.15.2).

For example, to define f:(x,y)→ xsin(y) and g(t):y→ f(t,y):
Input:

then:

Output:

Input:

Output:

For another example, suppose that you want to define the function h: (x,y) → [x*cos(y),x*sin(y)] and then you want to define the family of functions k(t) having t as parameter such that k(t)(y):=h(t,y). To define the function h(x,y)
Input:

To define properly the function k(t): Input:

then:

Output:

Input:

Output: