How does the function f(x)=0^{-x} behave for positive and complex numbers?

[zetafunction]

how this function f(x)= 0^{-x}

should be understood? , if x is NEGATIVE, we find no problems, since 0 raised to any power is 0

but how about x being a positive real number ? , or x being a PURE COMPLEX or complex number ?

could we consider a 'regularization' to this f(x) so f(x)_{reg}=0 is always 0

 

[Office_Shredder]

I would imagine the same way you would understand \sqrt{x}. It's assumed that the domain is restricted only to numbers that make sense.

If you have some context where you think you really need to plug 1 into the function, you should post the full setup here

 

[Petr Mugver]

In the real case, the function

f(x,y)=y^x=e^{x\log y}

is defined only for y>0

In the complex case, put

y=\rho e^{i\alpha}\qquad\textrm{and}\qquad x=a+ib

then

y^x=(\rho e^{i\alpha})^{(a+ib)}

and, after some calculations, you find

y^x=Re^{i\beta}

with

R=\rho^ae^{-\alpha b}\qquad\textrm{and}\qquad\beta=b\log\rho+\alpha a

So in the complex case you must have \rho\neq 0, which means y\neq 0.

I hope I didn't meke calculation errors...try it yourself!