Complex Exponents

Question

What does it mean to raise a number to a complex exponent, and why? A lot of the explanations that I've seen involve e, why is this?

I'm looking for an intuitive answer describing to how exponentiation works on a number plane (i.e. the complex number plane) rather than just a number line (the real numbers), where I understand exponentiation quite well. I'm not looking for a breakdown of Taylor series - I believe the proofs, I just don't grok them.

For example, if I was asking how sine worked, I'd want this, not this.

Unfortunately, if you got your first answer, it would not tell you anything about how the complex sine "works" (or is defined), while the latter answer would very naturally generalize to complex values. — Arturo Magidin, May 18 '11 at 18:08
Possible duplicate of http://math.stackexchange.com/questions/9770/understanding-imaginary-exponents/9777#9777 http://math.stackexchange.com/questions/9776/a-complex-number-to-the-power-of-another-complex-number — , May 18 '11 at 18:24
@Nate: please look at the accepted answer to the first question in Sivaram's comment. It may help with what you want. — Ross Millikan, May 18 '11 at 18:58
That definitely helped me understand why A^i = x+iy implies that (x,y) describes a point on the unit circle. However, it still doesn't help me understand how x and y are found, or why e plays such a big role. Can you give me some pointers to understanding those two things? — Nate, May 18 '11 at 19:02
@Nate: e plays a big role because $e^{ix}= \cos(x) + i \sin(x)$. So for $\phi \in \mathbb{R}$, $e^{i\phi}$ is a point on the unit circle in the complex plane. — Fabian, May 18 '11 at 19:05
See, that's exactly the part that I'm trying to understand by understanding complex exponents. Everybody just waves their hands and says "by definition" or pulls out a Taylor series, which is a proof, not a reason. I'm trying to figure out why $e^{ix}=cos(x) + i*sin(x)$. Lubos Motl's answer below helps, but if you have any extra pointers they'd be more than welcome. — Nate, May 18 '11 at 19:36
you can see the exponential $e^(ix)$ as the (local) inverse of the complex logz. In a sense, the complex log assigns to a number z its polar coordinate logz= |z|+iargz. If z is on the unit circle, its log is iargz (where arg depends on the branch). When the inverse is defined, log($e^z$)=$e^logz$=z, and then , while log assigns to z its polar coords., $e^z$ does the opposite; it describes the number which has a specific polar representation: the polar representation ix originates from the complex number cosx+isinx (this is subject to choice of branch) — gary, May 19 '11 at 06:14
There is no intuitive way of understanding the thing, and it's similar to the concept of the number $i$ itself. It will be helpful to look up the concept of analytic continuation. When you try to extend a "analytic" function (which almost every known function is)'s domain to a larger domain, there is a unique way of extending it. And such an extension is usually useful in the study of important problems (not only in math but in physics as well) despite the original intuitive meaning of the function being lost. The extension of real exponentation into complex domain is one of such example. — cr001, Jan 27 '19 at 08:16

score 12 · Accepted Answer · answered May 18 '11 at 19:03

Dear Nate, first, why is $e$ the preferred base for exponentials? Imagine that you have 1 dollar and your bank gives you 100% interest rate. After 1 year, you will have 2 dollars.

Now it offers you to add interests 100 times in a year but the interest is 1% at each moment. How much will you get? You will get $$ (1+0.01)^{100} \approx 2.704 $$ What if they add you $1/N \times $ 100% at $N$ moments of the year and you send $N$ to infinity? Well, you will have $e\approx 2.71828$ dollars after one year.

In fact, the general exponential - power with the base of $e$ - may be defined by this "repeated small interest" formula as $$ e^X = \exp(X) = \lim_{N\to \infty} \left(1+\frac {X}{N}\right)^N $$ It only has this simple form if the base is $e$. A more general power may be defined as $$ Y^X = \exp(X \ln Y) .$$ Here, $\ln$ is the natural logarithm so that $\exp\ln X = X$. If I replaced the base $e$ by another base such as $2$ or $10$, the "repeated small interest" formula above would have to contain $\ln 2$ or $\ln 10$ or other awkward factors at various places. It wouldn't be natural.

So instead of powers $Y^X$ and logarithms with general bases, you should think that in mathematics, only $\exp(X)$ and $\ln(X)$ are really needed, and all the other powers and logarithms may be expressed as composite functions. Also, $\exp(X)$ has the advantage that its derivative is exactly equal to the very same function $\exp(X)$. In particular, the derivative evaluated at $X=0$ is equal to one, very nice and simple. It would be $\ln(Y)$ if you used a different base $Y$ instead of $e$.

Now, what is the exponential of an imaginary exponent? Again, you may write $$\exp(iX) = \lim_{N\to\infty} \left(1+\frac {iX}{N}\right)^N $$ You multiply $N$ copies of a number that is very close to one. What do you get?

Well, the multiplication by a complex number has the effect of magnifying (or reducing) the plane, and rotating it. In particular, the absolute value of the number $(1+iX/N)$ is essentially one, up to second-order corrections that disappear in the $N\to \infty$ limit. So in the limit, $(1+iX/N)$ is effectively a number whose absolute value equals one.

Multiplying by complex numbers whose absolute value is equal to one looks like a rotation of the complex plane. The angles are preserved - those are some things one should know about the complex numbers. Moreover, it's clear that multiplying by $(1+iX/N)$ is equivalent to the rotation by $X/N$ in radians. If you multiply the same factor $N$ times, you simply get a rotation by $X/N$ in radians.

So the $N$th power of $1+iX/N$, in the limit $N\to\infty$, is the number that you get by rotating $1$ in the counter-clockwise direction by the angle $X$ in radians. Clearly, the answer is $$ \exp(iX) = \cos (X) + i \sin(X) $$ where the trigonometric functions have arguments in radians, of course. Once again, the mathematically natural unit of an angle is in radians for very similar reasons why the natural base of the powers or exponentials is $e$. Only in radians, it's true that the derivative of $\sin X$ equals $\cos X$ and many other things.

In fact, the previous formula makes it natural to say that $\cos X$ and $\sin X$ are not "independent" functions, either. They may be defined as $$\cos (X) = \frac 12 ( e^{iX} + e^{-iX} ) $$ $$\sin(X) = \frac{1}{2i} (e^{iX} - e^{-iX} ) $$ You may substitute the last two equations into the previous one or vice versa to check that everything is consistent.

Just to be sure, general complex numbers may also be exponentiated via $\exp(A+iB) = \exp(A)\exp(iB)$ where both factors are known.

Thanks, that clears a lot of things up. I still don't quite understand why multiplication by (1+iX/N) is equivalent to rotation by X/N - do you think you could clear that up a bit for me? — Nate, May 23 '11 at 19:40
@Nate Here is more detailed proof http://slesinsky.org/brian/misc/eulers_identity.html — uhbif19, Jun 20 '17 at 10:05

score 3 · Answer 2 · answered May 18 '11 at 18:46

There are various possible approaches, but I prefer to start with the exponential function $e^z$ as solution to the differential equation $f'(z) = f(z)$ with $f(0)=1$. The other exponentials are derived from this: $a^z = e^{z \log(a)}$ is a solution to $f'(z) = \log(a) f(z)$ with $f(0)=1$.

Complex Exponents

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