What Is the Discrepancy in the Value of ii in Euler's Formula?

[Dr. Seafood]

To compute this, we’ll make use of Euler’s formula cis(x) = e^ix = cos(x) + i·sin(x):

e^i(π/2) = cos(π/2) + i·sin(π/2) = i, and exponentiating by i we get
iⁱ = (e^i(π/2))ⁱ = e^i·i(π/2) = e^-π/2 ∈ ℝ.

But we also have cis(2πk + π/2) = i, k ∈ ℤ. Thus by the same logic, we get iⁱ = e^{-(2πk + π/2)} for k ∈ ℤ.

Infinitely many evaluations, so iⁱ doesn't have a distinct/unique value? This seems like a discrepancy. I don't have any strong arguments, but I have this W|A computation telling me that iⁱ is indeed distinct, since none of e^-5π/2, e^-9π/2, e^-13π/2, etc are equal.

Also, since the exponential function from ℝ onto ℝ⁺ is certainly injective, each of these numbers e^{-(2πk + π/2)} must be distinct. If iⁱ is distinct, why is its value e^-π/2 = 0.2078795... ?

Can anyone explain this?

 

[pmsrw3]

Well, there's nothing surprising about that. x^y is by definition e^{y log(x)}, and log is a multivalued function. Except for integer y, this always leads to multiple possible values of x^y. One usually resolves this by the somewhat artificial dodge of using the principal branch of log y, defined by convention, (just as $\sqrt{y}$ is taken to mean the positive square root of y, even though there is also a negative one).

 

[Dr. Seafood]

That's true, in that case I'll accept that iⁱ cannot be uniquely evaluated. But I've seen in other places (including on this forum) that iⁱ has a distinct value. This is clearly untrue by the logic in the OP post. What's the discrepancy?

 

[pmsrw3]

It does have a unique value -- if you choose the principal branch. (Which is to say, it has a unique value if you define it to.)

 

[micromass]

Indeed, the complex exponential is defined as

$$x^y=e^{yLog(x)}$$

The log is multivalued, so there will be an infinite number of possible $x^y$. But in many applications, we want a principal value of $x^y$. This is done by the formula

$$x^y=e^{yLog(x)}$$

But where the Log is now the principal branch and is single-valued. That is, we restricted the range of the Log such that it becomes single-valued.

This principal value of $x^y$ pops up in many places, for example in the definition of the Riemann-zeta function:

$$\sum{\frac{1}{n^s}}$$

where the s can be complex.