The Complex Exponential and Logarithm
We can now meet the most important holomorphic
function of all: the complex exponential. Using
Euler's formula,
we define e^z for z = x + iy by
e^z = e^{x + iy} = e^x\left(\cos y + i\sin y\right).
The real factor e^x sets the modulus, and the
(\cos y + i\sin y) factor is the unit-circle rotator that turns it
through angle y. So e^z has modulus
e^x and argument y.
What e^z keeps, and what it gains
It is entire — holomorphic on all of \mathbb{C},
with the clean derivative \frac{d}{dz}e^z = e^z, just like the real
case.
It still adds in the exponent. The law of exponents survives untouched:
e^{z + w} = e^z\,e^w.
But it is now periodic. Here is the genuine surprise. Because
\cos and \sin repeat every
2\pi, adding 2\pi i to
z changes nothing:
e^{z + 2\pi i} = e^z\,e^{2\pi i} = e^z\left(\cos 2\pi + i\sin 2\pi\right) = e^z \cdot 1 = e^z.
The complex exponential is periodic with period 2\pi i
— a property the real exponential, which is strictly increasing and one-to-one, never had. This
single fact is what makes the inverse interesting.
The logarithm is multivalued
The logarithm should invert the exponential: \log z is a
w with e^w = z. Writing
z = r e^{i\theta} and w = \log z:
\log z = \ln|z| + i\,\arg z = \ln r + i\theta.
Step 1 — recover the modulus. Since |e^w| = e^{\operatorname{Re} w}
must equal r, the real part is the ordinary real log,
\ln r.
Step 2 — recover the angle. The imaginary part is
\arg z — but the argument is only defined up to
2\pi, since e^z is periodic. So
\log z = \ln r + i(\theta + 2\pi k), \qquad k = 0, \pm 1, \pm 2, \dots
gives infinitely many values, stacked 2\pi apart up the
imaginary direction. The complex logarithm is multivalued.
The principal branch. To get a single function, pin the angle to the range
\arg z \in (-\pi, \pi]. This is the principal value
\operatorname{Log} z, with a branch cut along the
negative real axis (where the angle would have to jump from
+\pi to -\pi). Complex powers are then
built from it:
z^a = e^{a\,\operatorname{Log} z}.
Two famous values
Log(−1). The number -1 has modulus
1 and principal argument \pi, so
\operatorname{Log}(-1) = \ln 1 + i\pi = i\pi.
i^i is real. Here the multivalued machinery pays
off in a startling way. Use i^i = e^{i\,\operatorname{Log} i} with
\operatorname{Log} i = i\frac{\pi}{2} (modulus
1, argument \frac{\pi}{2}):
i^i = e^{\,i \cdot i\frac{\pi}{2}} = e^{\,i^2 \frac{\pi}{2}} = e^{-\pi/2} \approx 0.2079.
An imaginary number raised to an imaginary power is a perfectly ordinary real number.
For z = x + iy:
-
e^z = e^x(\cos y + i\sin y) is entire, obeys
e^{z+w} = e^z e^w, and is periodic with period
2\pi i;
-
its inverse \log z = \ln|z| + i\arg z is
multivalued (arg defined up to 2\pi); the
principal branch \operatorname{Log} takes
\arg z \in (-\pi, \pi], cut along the negative real axis;
-
powers are z^a = e^{a\,\operatorname{Log} z}; e.g.
\operatorname{Log}(-1) = i\pi and
i^i = e^{-\pi/2} (real!).
Imagine walking once anticlockwise around the origin. Your argument climbs steadily:
0, \tfrac{\pi}{2}, \pi, \tfrac{3\pi}{2}, 2\pi — and back where you
started, yet the logarithm has risen by 2\pi i. The function
cannot be both continuous and single-valued on a loop around 0.
The branch cut is the fence we install (conventionally along the negative
real axis) so you can never complete that loop without crossing it — restoring a single,
continuous \operatorname{Log} at the cost of a jump along the cut.
The origin itself, where this trouble concentrates, is a branch point.
The stacked values of log z
Move z around the unit circle with the angle slider (here
|z| = 1, so \ln|z| = 0 and every value
sits on the imaginary axis). The principal value
\operatorname{Log} z = i\theta is highlighted; the other branches
i(\theta + 2\pi k) march off above and below it, exactly
2\pi apart. That endless ladder is the multivaluedness.