How to condition arguments of a pure function?

Question

I would like to exclude the point {x=0,y=0} in the function definition

f = Function[{x, y}, {x/(x^2 + y^2), -(y/(x^2 + y^2))}]    

So far I tried ConditionalExpressionand /; without success.

Thanks!

Answer 1

As far as I know, you cannot have a pure function that holds its arguments depending on a condition on the arguments. Condition specifically works with pattern matching and pure functions don't use the pattern matcher. So the behaviour that f[0, 0] doesn't evaluate simply cannot be achieved with a pure function.

The alternative is some sort of default return value. Probably the easiest way to do this, is with something like If, Switch or Which:

f = Function[{x, y},
  If[ TrueQ[x == 0 || y == 0],
   Undefined,
   {x/(x^2 + y^2), -(y/(x^2 + y^2))}
   ]
 ];
f[0, 0]

Undefined

Note the use of TrueQ, which is necessary to deal with the case f[a, b] where a and b are symbols. The If will not evaluate otherwise, since a == 0 and b == 0 remain unevaluated (because they are equation, not truth statements). The test x === 0 is not useful here either, since 0. === 0 evaluates to False. PossibleZeroQ[x] is another test you could use in the If statement.