Defining algebraic functions

### 2.16.3 Defining algebraic functions

#### Defining a function from ℝp to ℝ

For p=1, e.g. for f : (x)→ x*sin(x), input :

`f(x):=x*sin(x)`

Or :

`f:=x->x*sin(x)`

Output :

`(x)->x*sin(x)`

If p>1, e.g. for f : (x,y)→ x*sin(y), input :

`f(x,y):=x*sin(y)`

Or :

`f:=(x,y)->x*sin(y)`

Output :

`(x,y)->x*sin(y)`

Warning !!! the expression after `->` is not evaluated. You should use `unapply` if you expect the second member to be evaluated before the function is defined.

#### Defining a function from ℝp to ℝq

For example:

- To define the function h : (x,y)→ (x*cos(y),x*sin(y)).
  Input :
  `h(x,y):=(x*cos(y),x*sin(y))`
  Output :
  `expr(" (x,y)->{
  x*cos(y),x*sin(y);
  }",0)`
- To 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)];}`
  Output :
  `(x,y)->{return([x*cos(y),x*sin(y)]);}`

Warning !!! the expression after `->` is not evaluated.

#### Defining families of function from ℝp−1 to ℝq using a function from ℝp to ℝq

Suppose that the function f: (x,y) → f(x,y) is defined, and we 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, we should not define g(t) by `g(t):=y->f(t,y)`, we have to use the `unapply` command.

For example, assuming that f:(x,y)→ xsin(y) and g(t): y→ f(t,y), input :

`f(x,y):=x*sin(y);g(t):=unapply(f(t,y),y)`

Output :

`((x,y)->x*sin(y), (t)->unapply(f(t,y),y))`

Input

`g(2)`

Output :

`y->2``·``sin(y)`

Input

`g(2)(1)`

Output :

`2``·``sin(1)`

Next example, suppose that the function h: (x,y) → [x*cos(y),x*sin(y)] is defined, and we 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 :

`h(x,y):=(x*cos(y),x*sin(y))`

To define properly the function k(t), input :

`k(t):=unapply(h(x,t),x)`

Output :

`(t)->unapply(h(x,t),x)`

Input

`k(2)`

Output :

`(x)->(x*cos(2),x*sin(2))`

Input

`k(2)(1)`

Output :

`(2*cos(1),2*sin(1))`