8.3.3  Defining algebraic functions

Defining a function from ℝ^p to ℝ^q.

If expr is an expression possibly involving a variable x, use it to define a function f either by

or

(see Section 3.4.1). Note that the expression after -> is not evaluated. You should use unapply (see Section 8.2.2) if you expect the second member to be evaluated before the function is defined.

Example

To define f(x)=xsin(x), input:

or:

Then:

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

To define f(x)=xsin(y), input:

or:

Then:

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)=(xcos(y),xsin(y)).

Then:

Define the function h(x,y)=[xcos(y),xsin(y)].

or:

or:

or:

or:

Then:

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 8.2.2).

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

then:

As another example, suppose that you want to define the function h: (x,y) → [xcos(y),xsin(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):

To define properly the function k(t):

then: