Complex logarithm

For each nonzero complex number ${\displaystyle z}$, the principal value ${\displaystyle \operatorname {Log} z}$ is the logarithm whose imaginary part lies in the interval ${\displaystyle (-\pi ,\pi ]}$.[2] The expression ${\displaystyle \operatorname {Log} 0}$ is left undefined since there is no complex number ${\displaystyle w}$ satisfying ${\displaystyle e^{w}=0}$.[1]

When the notation ${\displaystyle \log z}$ appears without any particular logarithm having been specified, it is generally best to assume that the principal value is intended. In particular, this gives a value consistent with the real value of ${\displaystyle \ln z}$ when ${\displaystyle z}$ is a positive real number. The capitalization in the notation ${\displaystyle {\text{Log}}}$ is used by some authors[2] to distinguish the principal value from other logarithms of ${\displaystyle z}$.

The polar form of a nonzero complex number ${\displaystyle z=x+yi}$ is ${\displaystyle z=re^{i\theta }}$, where ${\textstyle r=|z|={\sqrt {x^{2}+y^{2}}}}$ is the absolute value of ${\displaystyle z}$, and ${\displaystyle \theta }$ is its argument. The absolute value is real and positive. The argument is defined up to addition of an integer multiple of 2π. Its principal value is the value that belongs to the interval ${\displaystyle (-\pi ,\pi ]}$, which is expressed as ${\displaystyle \operatorname {atan2} (y,x)}$.

This leads to the following formula for the principal value of the complex logarithm:

${\displaystyle \operatorname {Log} z=\ln r+i\theta =\ln |z|+i\operatorname {Arg} z=\ln {\sqrt {x^{2}+y^{2}}}+i\operatorname {atan2} (y,x).}$

For example, ${\displaystyle \operatorname {Log} (-3i)=\ln 3-\pi i/2}$, and ${\displaystyle \operatorname {Log} (-3)=\ln 3+\pi i}$.

Another way to describe ${\displaystyle \operatorname {Log} z}$ is as the inverse of a restriction of the complex exponential function, as in the previous section. The horizontal strip ${\displaystyle S}$ consisting of complex numbers ${\displaystyle w=x+yi}$ such that ${\displaystyle -\pi <y\leq \pi }$ is an example of a region not containing any two numbers differing by an integer multiple of ${\displaystyle 2\pi i}$, so the restriction of the exponential function to ${\displaystyle S}$ has an inverse. In fact, the exponential function maps ${\displaystyle S}$ bijectively to the punctured complex plane ${\displaystyle \mathbb {C} ^{*}=\mathbb {C} \setminus \{0\}}$, and the inverse of this restriction is ${\displaystyle \operatorname {Log} \colon \mathbb {C} ^{*}\to S}$. The conformal mapping section below explains the geometric properties of this map in more detail.

On the region ${\displaystyle \mathbb {C} -\mathbb {R} _{\leq 0}}$ consisting of complex numbers that are not negative real numbers or 0, the function ${\displaystyle \operatorname {Log} z}$ is the analytic continuation of the natural logarithm. The values on the negative real line can be obtained as limits of values at nearby complex numbers with positive imaginary parts.

Not all identities satisfied by ${\displaystyle \ln }$ extend to complex numbers. It is true that ${\displaystyle e^{\operatorname {Log} z}=z}$ for all ${\displaystyle z\not =0}$ (this is what it means for ${\displaystyle \operatorname {Log} z}$ to be a logarithm of ${\displaystyle z}$), but the identity ${\displaystyle \operatorname {Log} (e^{z})=z}$ fails for ${\displaystyle z}$ outside the strip ${\displaystyle S}$. For this reason, one cannot always apply ${\displaystyle {\text{Log}}}$ to both sides of an identity ${\displaystyle e^{z}=e^{w}}$ to deduce ${\displaystyle z=w}$. Also, the identity ${\displaystyle \operatorname {Log} (z_{1}z_{2})=\operatorname {Log} z_{1}+\operatorname {Log} z_{2}}$ can fail: the two sides can differ by an integer multiple of ${\displaystyle 2\pi i}$;[1] for instance,

${\displaystyle \operatorname {Log} ((-1)i)=\operatorname {Log} (-i)=\ln(1)-{\frac {\pi i}{2}}=-{\frac {\pi i}{2}},}$

but

${\displaystyle \operatorname {Log} (-1)+\operatorname {Log} (i)=\left(\ln(1)+\pi i\right)+\left(\ln(1)+{\frac {\pi i}{2}}\right)={\frac {3\pi i}{2}}\neq -{\frac {\pi i}{2}}.}$

The function ${\displaystyle \operatorname {Log} z}$ is discontinuous at each negative real number, but continuous everywhere else in ${\displaystyle \mathbb {C} ^{*}}$. To explain the discontinuity, consider what happens to ${\displaystyle \arg z}$ as ${\displaystyle z}$ approaches a negative real number ${\displaystyle a}$. If ${\displaystyle z}$ approaches ${\displaystyle a}$ from above, then ${\displaystyle \arg z}$ approaches ${\displaystyle \pi }$, which is also the value of ${\displaystyle \arg a}$ itself. But if ${\displaystyle z}$ approaches ${\displaystyle a}$ from below, then ${\displaystyle \arg z}$ approaches ${\displaystyle -\pi }$. So ${\displaystyle \arg z}$ "jumps" by ${\displaystyle 2\pi }$ as ${\displaystyle z}$ crosses the negative real axis, and similarly ${\displaystyle \operatorname {Log} z}$ jumps by ${\displaystyle 2\pi i}$.

The argument above involving the unit circle generalizes to show that no branch of ${\displaystyle \log z}$ exists on an open set ${\displaystyle U}$ containing a closed curve that winds around 0. One says that "${\displaystyle \log z}$ has a branch point at 0". To avoid containing closed curves winding around 0, ${\displaystyle U}$ is typically chosen as the complement of a ray or curve in the complex plane going from 0 (inclusive) to infinity in some direction. In this case, the curve is known as a branch cut. For example, the principal branch has a branch cut along the negative real axis.

If the function ${\displaystyle \operatorname {L} (z)}$ is extended to be defined at a point of the branch cut, it will necessarily be discontinuous there; at best it will be continuous "on one side", like ${\displaystyle \operatorname {Log} z}$ at a negative real number.

Each branch ${\displaystyle \operatorname {L} (z)}$ of ${\displaystyle \log z}$ on an open set ${\displaystyle U}$ is the inverse of a restriction of the exponential function, namely the restriction to the image ${\displaystyle \operatorname {L} (U)}$. Since the exponential function is holomorphic (that is, complex differentiable) with nonvanishing derivative, the complex analogue of the inverse function theorem applies. It shows that ${\displaystyle \operatorname {L} (z)}$ is holomorphic on ${\displaystyle U}$, and ${\displaystyle \operatorname {L} '(z)=1/z}$ for each ${\displaystyle z}$ in ${\displaystyle U}$.[2] Another way to prove this is to check the Cauchy–Riemann equations in polar coordinates.[2]

The function ${\displaystyle \ln(x)}$ for real ${\displaystyle x>0}$ can be constructed by the formula

$${\displaystyle \ln(x)=\int _{1}^{x}{\frac {du}{u}}.}$$

If the range of integration started at a positive number ${\displaystyle a}$ other than 1, the formula would have to be

$${\displaystyle \ln(x)=\ln(a)+\int _{a}^{x}{\frac {du}{u}}}$$

instead.

In developing the analogue for the complex logarithm, there is an additional complication: the definition of the complex integral requires a choice of path. Fortunately, if the integrand is holomorphic, then the value of the integral is unchanged by deforming the path (while holding the endpoints fixed), and in a simply connected region ${\displaystyle U}$ (a region with "no holes"), any path from ${\displaystyle a}$ to ${\displaystyle z}$ inside ${\displaystyle U}$ can be continuously deformed inside ${\displaystyle U}$ into any other. All this leads to the following:

If ${\displaystyle U}$ is a simply connected open subset of ${\displaystyle \mathbb {C} }$ not containing 0, then a branch of ${\displaystyle \log z}$ defined on ${\displaystyle U}$ can be constructed by choosing a starting point ${\displaystyle a}$ in ${\displaystyle U}$, choosing a logarithm ${\displaystyle b}$ of ${\displaystyle a}$, and defining

$${\displaystyle L(z)=b+\int _{a}^{z}{\frac {dw}{w}}}$$

for each ${\displaystyle z}$ in ${\displaystyle U}$.[5]

The various branches of ${\displaystyle \log z}$ cannot be glued to give a single continuous function ${\displaystyle \log \colon \mathbb {C} ^{*}\to \mathbb {C} }$ because two branches may give different values at a point where both are defined. Compare, for example, the principal branch ${\displaystyle \operatorname {Log} z}$ on ${\displaystyle \mathbb {C} -\mathbb {R} _{\leq 0}}$ with imaginary part ${\displaystyle \theta }$ in ${\displaystyle (-\pi ,\pi )}$ and the branch ${\displaystyle \operatorname {L} (z)}$ on ${\displaystyle \mathbb {C} -\mathbb {R} _{\geq 0}}$ whose imaginary part ${\displaystyle \theta }$ lies in ${\displaystyle (0,2\pi )}$. These agree on the upper half plane, but not on the lower half plane. So it makes sense to glue the domains of these branches only along the copies of the upper half plane. The resulting glued domain is connected, but it has two copies of the lower half plane. Those two copies can be visualized as two levels of a parking garage, and one can get from the ${\displaystyle {\text{Log}}}$ level of the lower half plane up to the ${\displaystyle {\text{L}}}$ level of the lower half plane by going ${\displaystyle 2\pi }$ radians counterclockwise around 0, first crossing the positive real axis (of the ${\displaystyle {\text{Log}}}$ level) into the shared copy of the upper half plane and then crossing the negative real axis (of the ${\displaystyle {\text{L}}}$ level) into the ${\displaystyle {\text{L}}}$ level of the lower half plane.

One can continue by gluing branches with imaginary part ${\displaystyle \theta }$ in ${\displaystyle (\pi ,3\pi )}$, in ${\displaystyle (2\pi ,4\pi )}$, and so on, and in the other direction, branches with imaginary part ${\displaystyle \theta }$ in ${\displaystyle (-2\pi ,0)}$, in ${\displaystyle (-3\pi ,-\pi )}$, and so on. The final result is a connected surface that can be viewed as a spiraling parking garage with infinitely many levels extending both upward and downward. This is the Riemann surface ${\displaystyle R}$ associated to ${\displaystyle \log z}$.[7]

A point on ${\displaystyle R}$ can be thought of as a pair ${\displaystyle (z,\theta )}$ where ${\displaystyle \theta }$ is a possible value of the argument of ${\displaystyle z}$. In this way, R can be embedded in ${\displaystyle \mathbb {C} \times \mathbb {R} \approx \mathbb {R} ^{3}}$.

Because the domains of the branches were glued only along open sets where their values agreed, the branches glue to give a single well-defined function ${\displaystyle \log _{R}\colon R\to \mathbb {C} }$.[8] It maps each point ${\displaystyle (z,\theta )}$ on ${\displaystyle R}$ to ${\displaystyle \ln |z|+i\theta }$. This process of extending the original branch ${\displaystyle {\text{Log}}}$ by gluing compatible holomorphic functions is known as analytic continuation.

There is a "projection map" from ${\displaystyle R}$ down to ${\displaystyle \mathbb {C} ^{*}}$ that "flattens" the spiral, sending ${\displaystyle (z,\theta )}$ to ${\displaystyle z}$. For any ${\displaystyle z\in \mathbb {C} ^{*}}$, if one takes all the points ${\displaystyle (z,\theta )}$ of ${\displaystyle R}$ lying "directly above" ${\displaystyle z}$ and evaluates ${\displaystyle \log _{R}}$ at all these points, one gets all the logarithms of ${\displaystyle z}$.

Instead of gluing only the branches chosen above, one can start with all branches of ${\displaystyle \log z}$, and simultaneously glue every pair of branches ${\displaystyle L_{1}\colon U_{1}\to \mathbb {C} }$ and ${\displaystyle L_{2}\colon U_{2}\to \mathbb {C} }$ along the largest open subset of ${\displaystyle U_{1}\cap U_{2}}$ on which ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$ agree. This yields the same Riemann surface ${\displaystyle R}$ and function ${\displaystyle \log _{R}}$ as before. This approach, although slightly harder to visualize, is more natural in that it does not require selecting any particular branches.

If ${\displaystyle U'}$ is an open subset of ${\displaystyle R}$ projecting bijectively to its image ${\displaystyle U}$ in ${\displaystyle \mathbb {C} ^{*}}$, then the restriction of ${\displaystyle \log _{R}}$ to ${\displaystyle U'}$ corresponds to a branch of ${\displaystyle \log z}$ defined on ${\displaystyle U}$. Every branch of ${\displaystyle \log z}$ arises in this way.

The projection map ${\displaystyle R\to \mathbb {C} ^{*}}$ realizes ${\displaystyle R}$ as a covering space of ${\displaystyle \mathbb {C} ^{*}}$. In fact, it is a Galois covering with deck transformation group isomorphic to ${\displaystyle \mathbb {Z} }$, generated by the homeomorphism sending ${\displaystyle (z,\theta )}$ to ${\displaystyle (z,\theta +2\pi )}$.

As a complex manifold, ${\displaystyle R}$ is biholomorphic with ${\displaystyle \mathbb {C} }$ via ${\displaystyle \log _{R}}$. (The inverse map sends ${\displaystyle z}$ to ${\displaystyle \left(e^{z},\operatorname {Im} (z)\right)}$.) This shows that ${\displaystyle R}$ is simply connected, so ${\displaystyle R}$ is the universal cover of ${\displaystyle \mathbb {C} ^{*}}$.

Just as for real numbers, one can define for complex numbers ${\displaystyle b}$ and ${\displaystyle x}$

${\displaystyle \log _{b}x={\frac {\log x}{\log b}},}$

with the only caveat that its value depends on the choice of a branch of log defined at ${\displaystyle b}$ and ${\displaystyle x}$ (with ${\displaystyle \log b\not =0}$). For example, using the principal value gives

${\displaystyle \log _{i}e={\frac {\operatorname {Log} e}{\operatorname {Log} i}}={\frac {1}{\pi i/2}}=-{\frac {2i}{\pi }}.}$

If f is a holomorphic function on a connected open subset ${\displaystyle U}$ of ${\displaystyle \mathbb {C} }$, then a branch of ${\displaystyle \log f}$ on ${\displaystyle U}$ is a continuous function ${\displaystyle g}$ on ${\displaystyle U}$ such that ${\displaystyle e^{g(z)}=f(z)}$ for all ${\displaystyle z}$ in ${\displaystyle U}$. Such a function ${\displaystyle g}$ is necessarily holomorphic with ${\displaystyle g'(z)=f'(z)/f(z)}$ for all ${\displaystyle z}$ in ${\displaystyle U}$.

If ${\displaystyle U}$ is a simply connected open subset of ${\displaystyle \mathbb {C} }$, and ${\displaystyle f}$ is a nowhere-vanishing holomorphic function on ${\displaystyle U}$, then a branch of ${\displaystyle \log f}$ defined on ${\displaystyle U}$ can be constructed by choosing a starting point a in ${\displaystyle U}$, choosing a logarithm ${\displaystyle b}$ of ${\displaystyle f(a)}$, and defining

${\displaystyle g(z)=b+\int _{a}^{z}{\frac {f'(w)}{f(w)}}\,dw}$

for each ${\displaystyle z}$ in ${\displaystyle U}$.[2]