The equation in question is: . It would be interesting to find a constant point in a function like exponent. Obviously there are no real solutions. But what if we consider complex numbers, are there any constant exponents?
We are looking for a complex number such as . Let , where - real part, - imaginary part. Following the expanded definition of the exponent:
Thus we get a system of two equations:
Since there are no real solutions, which also means . That means we can safely divide by and express :
Using this expression we get an equation that should help us get the value of :
Unfortunately, I wasn't able to derive a concrete root of this equation. However, let's consider these two functions:
Both are even functions. If we build their graphs we get:
- green, - red)(
As we can see there is an infinite number of intersections occurring roughly once in every interval. That means there is an infinite number of fixed complex points who are equal to their exponent which I find fascinating.
If you have any idea how to derive or found an error, feel free to comment.