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

In summary, the function f(x) = 0^(-x) raises the question of how it should be understood when x is a positive real number or a pure or complex number. A possible solution could be to redefine the function as f(x)_reg = 0, ensuring that it always equals 0. However, in the real case, the function is only defined for y>0 and in the complex case, it requires y to be non-zero. Therefore, it is assumed that the domain is restricted to numbers that make sense. If there is a need to plug in 1 for the function, the full setup must be provided.
  • #1
zetafunction
391
0
how this function [tex] f(x)= 0^{-x} [/tex]

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 [tex] f(x)_{reg}=0 [/tex] is always 0
 
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  • #2
I would imagine the same way you would understand [tex]\sqrt{x}[/tex]. 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
 
  • #3
In the real case, the function

[tex]f(x,y)=y^x=e^{x\log y}[/tex]

is defined only for [tex]y>0[/tex]

In the complex case, put

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

then

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

and, after some calculations, you find

[tex]y^x=Re^{i\beta}[/tex]

with

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

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

I hope I didn't meke calculation errors...try it yourself! :redface:
 
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1. What is a "strange function"?

A strange function is a mathematical function that exhibits unusual or unexpected behavior. It may have properties that are not typically seen in other functions, such as discontinuities, infinite or undefined values, or strange patterns in its graph.

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A "strange function" differs from a regular function in that it does not follow the usual rules and patterns that we expect from mathematical functions. It may have properties that are not typically seen in other functions, such as discontinuities, infinite or undefined values, or strange patterns in its graph.

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While "strange functions" may not have direct practical applications, they are often studied and used in theoretical mathematics to better understand the behavior and limits of different types of functions. They can also be used to create interesting and challenging problems for students to solve.

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