I do not understand. You want an analytic function so that and for all real . Is that what you are asking? Because the word "harmonic" is strange here.
Try . This is not everywhere analytic but it almost works.
I've been in trouble enough lately, I figured I would get up to speed on a few things. Pondering elementary Complex Analysis, I made it only to page 53 before I managed to get stuck.
I want a non-zero Harmonic Function that disappears on the hyperbola y = 1/x.
I'm staring at f(z) = z^2.
I created a lovely page (or 3) of algebraic manipulations that lead to interesting results and relationships, but don't solve the problem at hand. I figure I'm just missing something.
Do I get a hint? What about f(z) = z^2 is supposed to impress me?
Right. I seem already to have picked up the somewhat cheesy vocabulary of the author. Maybe I need a new book. Also, it seems I am assuming you can see what I'm seeing. How many students have I told not to do that?!
In any case:
1) It's u(x,y) on
2) "disappears" or "vanishes" seems to mean "u(x,y)= 0"
So u(x,1/x) = 0, u(x,y) is not constant, and u(x,y) satisfies Laplace's Equation, second partial derivatives sum to zero.
I have a hint, "Look at f(z) = z^2". Sadly, this "look" doesn't seem to be clarifying anything.