Functional equation with two inputs: $f(x,y)f(y,x)=1$

Question

Some context, I was trying to derive something and stumbled upon $f(x,y)f(y,x)=1$, when $x , y \ne 0$. Could someone suggest some tricks or books that I could apply to these types of questions? By the way, I'm trying to solve for $f(x,y)$ and I'm aware that one solution could be $f(x,y)=\frac xy$, but I would like to know how to derive it.

Thank you.

Kavi Rama Murthy · Answer 1 · 2021-10-08T23:12:30.567

3

There are infinitely many functions satisfying this property. You can start with any function $f(x,y)$ defined for $\{(x,y): x \leq y\}$ satisfying $f(x,x)=\pm 1$ which does not vanish at any point and define $f(y,x)=\frac 1 {f(x,y)}$ for $x >y$.

edited Oct 08 '21 at 23:12

answered Jul 05 '19 at 23:30

Kavi Rama Murthy

311,013

Would there be a book you could refer me to? Specifically, any one that deals with functionals with more than one input? – BonInSossusvlei Jul 05 '19 at 23:43
You also need $f(x,x) \in \pm 1$ $\forall x $ though. – Mike Oct 08 '21 at 22:01
@Mike You are right. Thanks for pointing out. – Kavi Rama Murthy Oct 08 '21 at 23:13

Functional equation with two inputs: $f(x,y)f(y,x)=1$

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