What is 0^{i}?

garnentas3m

garnentas3m

Answered question

2022-01-15

What is 0i?

Answer & Explanation

Bubich13

Bubich13

Beginner2022-01-16Added 36 answers

There is no such thing as 0i encompassed by a coherent definition of complex powers. For any rational p it is reasonable to define 0p=0. This is related to the fact that for a fixed z,zp has finitely many values, and is borne out in the fact that the Riemann surface for zp includes a point labeled 0.
However, a power with nonreal exponent has infinitely many values. This arises from arbitrary powers being properly defined via the logarithm. The Riemann surface for log-a lovely thing often viewed as a helix, and biholomorphic with the complex plane-does not admit smoothly incorporating a point labeled 0. This omission is intrinsic to the logarithm, and so precludes any meaningful value of 0i.
A formula for the sometimes infinite lattice of values of zp is
zp=exp(p(ln|z|+i arg z+i2πk))(kZ)
In particular
zi=exp(iln|z|)earg z2πk(kZ)
It is not hard to see that with z in an arbitrarily small disk about 0, any complex value whatsoever, other than 0, can be obtained as a value of zi. So there is no hope of any smoothness or regularity in assigning a value to 0i.
zurilomk4

zurilomk4

Beginner2022-01-17Added 35 answers

This value 0i is usually left undefined. But if to define it should be seemingly the most naturally assigned the value 1.
Exponential function az has the period i|2πlog(a)|. When a comes to 0, the period also comes to 0. This means the value in i should be the same as in 0. In other words, 0ix=00=1 for any real x.

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