Understanding imaginary exponents
Greetings!
I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean?
I've read a few pages on this issue, and they all seem to boil down to the same thing:
- Any real number $x$ can be written as $e^{\ln{x}}$ (seems obvious enough.)
- Mumble mumble mumble
- This is equivalent to $e^{\cos{x} + i\sin{x}}$
Clearly I'm missing something in step 2. I understand (at least I think I do) how the complex number $\cos{x} + i\sin{x}$ maps to a point on the unit circle in a complex plane.
What I am missing, I suppose, is how this point is related to the natural log of $x$. Moreover, I don't understand what complex exponentiation is. I can understand integer exponentiation as simple repeated multiplication, and I can understand other things (like fractional or negative exponents) by analogy with the operations that undo them. But what does it mean to repeat something $i$ times?
$\endgroup$ 76 Answers
$\begingroup$Consider a real number $A$, and take it to the power $i$. If our system of complex numbers is to be consistent, then $A^i$ must be a complex number; in other words, there must be two real numbers $x$ and $y$, which depend on $A$, such that:
$A^i=x+iy$
Furthermore, we can write $A^{-i}=x-iy$ for the same $x$ and $y$. Hence:
$x^2+y^2=(x+iy)(x-iy)=A^iA^{-i}=A^{i-i}=A^0=1$
We have shown that for any real number $A$, $|A^i|=1$, and therefore $A^i$ corresponds to a complex number which lies some angle $\theta$ along the unit circle.
Now consider the sine and cosine functions for extremely small angles $\epsilon$. A tiny angle $\epsilon$ cuts out a slice of the unit circle, and the curvature of the circumference over this small angle is negligible. We can therefore think of this slice as a right triangle with angle $\epsilon$, and the hypotenuse and adjacent sides are both length one since they correspond to the radius of the unit circle.
Using the formula for the arc length of a circle, it's easy to determine that in the right triangle formed by the small angle approximation, the length of the side opposite to the angle $\epsilon$ is equal to $\epsilon$. We can read off the $(x,y)$ coordinates from this diagram (which are $(cos(\epsilon),sin(\epsilon))$), and therefore we conclude that for very small angles $\epsilon$:
$sin(\epsilon) \approx \epsilon \hspace{10mm} cos(\epsilon) \approx 1$
therefore $cos(\epsilon) + isin(\epsilon) \approx 1+i\epsilon$, and hence for real numbers $A$ which are extremely close to one (so that $lnA$ is small), the complex number $A^i$ lies approximately at an angle $lnA$ along the unit circle, since $A^i=e^{ilnA}\approx 1+i(lnA)$.
$\endgroup$ 9 $\begingroup$The complex exponential $e^z$ for complex $z=x+iy$ preserves the law of exponents of the real exponential and satisfies $e^0=1$.
By definition
$$e^z=e^{x+iy}=e^xe^{iy}=e^x(\cos y+i\sin y)$$
which agrees with the real exponential function when $y=0$.
The principal logarithm of $z=x+iy$ is the complex number
$$w=\text{Log }z=\log |z|+i\arg z$$
so that $e^w=z$, where $\arg z$ (the principal argument of $z$) is the real number in $-\pi\lt \arg z\le \pi$, with $x=|z|\cos (\arg z)$ and $y=|z|\sin (\arg z)$.
The complex power is
$$z^w=e^{w\text{ Log} z}.$$
In your example $z=x,w=i$ is therefore $x^i=e^{i \text{ Log}x}$.
If $x>0$, $\text{Log }x=\log x$. If $x<0$, $\text{Log }x=\log |x|+i\pi$.
Examples:
$(-1)^i=e^{i\text{Log }(-1)}=e^{i(i\pi)}=e^{-\pi}$.
$2^i=e^{i\text{Log }(2)}=e^{i\log 2}=\cos (\log 2)+i\sin (\log 2)$.
$(-2)^i=e^{i\text{Log }(-2)}=e^{i(\log 2+i\pi)}=e^{i\log 2}e^{-\pi}=(\cos (\log 2)+i\sin (\log 2))e^{-\pi}.$
$\endgroup$ 2 $\begingroup$Lets say you want to figure out what $x^{a + ib}$ is, then like you mentioned, you start by writing
$x^{a + ib} = e^{a \ln(x) + i b \ln(x)}$
This can be split up though as
$ e^{a \ln(x) + i b \ln(x)} = e^{a \ln(x)} e^{i b \ln(x)} = x^a e^{i b \ln(x)} $
and then you can use euler's formula to take care of the imaginary exponent, so that
$x^{a + ib} = x^a \cos(b \ln(x)) + i x^a \sin(b \ln(x)) $
This formula does not provide much intuition as to what is really happening though. You mentioned that you can understand integer exponentiation as simple repeated multiplication, but I don't think that that is the right way to look at it in complex analysis. I think it is much better to look at exponentiation geometrically.
When exponentiating a real number by a complex value, we found that
$x^{a + ib} = x^a e^{i b \ln(x)} $
so exponentiating $x$ by $a + ib$ gives you the point in the plane with magnitude $x^a$ and angle $b \ln(x)$.
$\endgroup$ 2 $\begingroup$It is a way to extend your real variable function into the complex domain in a "nice" (read nice as holomorphic) way. You might ask the question why do we need to even extend in the first place? The need for extension comes from the fact that a $n^{th}$ order polynomial with real co-efficients need not have all its roots as real numbers. So you end up with these complex numbers. Once we find that we have these new numbers, we begin to wonder why should we not try to feed these numbers into functions which are defined on the real variables. But then we find that we have a zillion ways to define these functions for these new complex numbers. A "nice" way to extend these functions is to do in a way that the extended function is holomorphic. That is why you extend $e^{i \theta}$ to the complex plane as $\cos(\theta) + i \sin(\theta)$. Once you do this extension then all the other things related to logarithms and exponentiation follows in a logically coherent way. For instance, we can now define $x^i$ (say $x>0$ or else rewrite $x$ as $-y$ where $y>0$ and write $-1$ as $e^{i \pi}$) as $(e^{\log (x)})^{i} = e^{i \log(x)} = \cos(\log(x)) + i \sin(\log(x))$
This is how I understand it.
$\endgroup$ 2 $\begingroup$The quantity $x^i$ is well-defined if $x$ is a positive real number. For negative (or complex) values of $x$, the evaluation is open to multiple interpretations.
You have the following. The natural logarithm of positive real numbers is well-defined, so:
$x^i = \exp(i\cdot\ln(x))$
Bear in mind that for real $x > 0$, all one knows about $\ln(x)$ is that it is real. It might be positive, negative, or zero depending on the comparison of $x$ to $1$.
Now the function $\exp$ can be defined by a power series that converges everywhere in the complex plane. As it turns out, that definition of the exponential function implies the following when the argument is purely imaginary (as here):
$x^i = \exp(i\cdot\ln(x)) = \cos(\ln(x)) + i\cdot\sin(\ln(x))$
For more see:
[Euler's formula -- Wikipedia]
$\endgroup$ 3 $\begingroup$First think about what you are really doing with exponents.
If you square a value, you are saying output is from two identical constraints. Like a 45 degree right triangle. The hypotenuse changes based on both sides growing or shrinking the exact same way. That's not to say whatever is doing the squaring is a triangle. It just acts that way. Same with cubing a value. Output is from three identical constraints, like how a surface arc on a sphere changes based on three radii growing or shrinking the exact same way.
Imaginary exponents are just the same. i, 2i, 3i are just like 1, 2 ,3: identity, square, and cube. They just need to run into another imaginary exponent to manifest their value, or you are carrying a lot of extra stuff you don't see.
The reason we like e is because it's derivative is the same as it's function's output. Put in 0, you get 1, and your slope is 1. Now you can start (real,imaginary) at (1,0), and go smoothly to (0,1) without messing with the constants.
This started with Gauss btw. His school teacher asked him to sum the values from 0 to 100, and he saw that as a field of 100, (100+0, 99+1, 98+2 ...)
Some real world examples:
You can harvest an apple orchard by picking the apple from the stem, or cutting the branch. If you cut the branch, you will cut apple blossoms. The apple blossoms are like an imaginary number, and you could make a time based imaginary function that steps out real world apples from the imaginary apples in the blossoms.
Alternating current works by turning off the power in the line intermittently to save power. All they do is switch terminals at the alternator, very fast, 60 times a second, meaning it adds up a lot, with out of phase equipment costing millions more in use. It's like hitting a pendulum on the upswing vs. the downswing. The real value is the 110v you circuit activates at. The imaginary value is whether you are on the upswing and downswing, and if you get a little more, or if you spend money fighting against the phase to close the circuit on the neutral wire.
Stress in odd shape parts with complex loading, like a hip replacement, can flow through with balance, canceling each other out and not stressing the part. Or they can act against each other and build up a lot of internal stress, which is like an imaginary number. Load in the right direction, no matter the magnitude, and all that stress is released, so you're not pressing on metal but a spring releasing away from you dragging you down.
$\endgroup$
