Complex Logarithm

Definition

1.The real exponential is a bijection from $\mathbb{R}$ to $(0,\infty),$ and its inverse function is the real logarithm. The complex exponential has period $2\pi i,$ so it is not injective on $\mathbb{C}.$ For every $w\in\mathbb{C}$ and every integer $k\in\mathbb{Z},$ we have:

$$e^{w+2k\pi i}=e^w$$

2.Two complex numbers that differ by an integer multiple of $2\pi i$ have the same exponential. Fix a nonzero complex number $z.$ Every solution $w\in\mathbb{C}$ of the following equation is called a complex logarithm of $z:$

$$e^w=z$$

3.To solve this equation, write $w=u+iv$ with $u,v\in\mathbb{R},$ and write $z$ in exponential form as $z=re^{i\theta},$ where $r=|z|>0$ and $\theta$ is one argument of $z.$ The exponential of $w$ is:

$$e^w=e^{u+iv}=e^u(\cos v+i\sin v)$$

4.The equality $e^w=z$ holds exactly when $e^u=r$ and $v=\theta+2k\pi$ for some $k\in\mathbb{Z}.$ Since $e^u=r,$ we have $u=\ln r.$ The complete set of logarithms of $z$ is:

$$\log z:=\{\ \ln|z|+i(\theta+2k\pi)\mid k\in\mathbb{Z}\ \}$$

5.Choosing another argument $\theta+2m\pi$ does not change this set, because the index $k$ then shifts by the integer $m.$ All logarithms of a fixed $z$ have real part $\ln|z|,$ and their imaginary parts differ by integer multiples of $2\pi.$ In the complex plane they lie on the vertical line whose real coordinate is $\ln|z|,$ equally spaced at distance $2\pi.$

6.For every finite $u,$ the modulus $|e^{u+iv}|=e^u$ is positive. The complex exponential never vanishes, so the number $0$ has no complex logarithm.

The principal logarithm

7.A standard single-valued choice uses the principal argument. For $z\neq0,$ the number $\mathrm{Arg}(z)$ is the unique argument in the interval $(-\pi,\pi].$ The principal logarithm is:

$$\mathrm{Log}(z):=\ln|z|+i\mathrm{Arg}(z)$$

8.The capitalized notation $\mathrm{Log}$ is the selected value, while the lowercase notation $\log z$ is the full set. On the positive real axis the definition agrees with the real natural logarithm. If $x>0,$ then $\mathrm{Arg}(x)=0$ and $\mathrm{Arg}(-x)=\pi,$ so the corresponding principal values are:

$$\mathrm{Log}(x)=\ln x,\qquad \mathrm{Log}(-x)=\ln x+i\pi$$

9.The number $z=-\sqrt{3}-i$ is in the third quadrant. Its modulus and principal argument are:

$$ \begin{align} |z|&=\sqrt{(-\sqrt{3})^2+(-1)^2}=2\\[6pt] \mathrm{Arg}(z)&=-\frac{5\pi}{6} \end{align} $$

10.Its principal logarithm is:

$$\mathrm{Log}(-\sqrt{3}-i)=\ln 2-\frac{5\pi i}{6}$$

11.All logarithms of $-\sqrt{3}-i$ are:

$$\log(-\sqrt{3}-i)=\left\{\ \ln 2+i\left(-\frac{5\pi}{6}+2k\pi\right)\mid k\in\mathbb{Z}\ \right\}$$

12.The value for $k=0$ is principal, and every other value differs from it by $2k\pi i.$

13.The principal logarithm is a right inverse of the exponential on nonzero complex numbers:

$$e^{\mathrm{Log}(z)}=z$$

14.The inverse identity in the other order holds only on a horizontal strip. If $w=x+iy,$ let $k$ be the unique integer for which $-\pi

$$\mathrm{Log}(e^w)=w-2k\pi i$$

15.The equality $\mathrm{Log}(e^w)=w$ holds exactly when $-\pi<\mathrm{Im}(w)\leq\pi.$ On this half-open horizontal strip, the principal logarithm is the inverse of the exponential.

Logarithmic branches and branch cuts

16.Let $U$ be a connected open subset of $\mathbb{C}^{*}:=\mathbb{C}\setminus\{\ 0\ \}.$ A logarithmic branch on $U$ is a continuous function $L:U\to\mathbb{C}$ such that the following identity holds for every $z\in U:$

$$e^{L(z)}=z$$

17.No logarithmic branch exists on the whole punctured plane $\mathbb{C}^{*}.$ If such a branch $L$ existed, the parametrization of the unit circle by $z=e^{it},$ with $0\leq t\leq2\pi,$ would define the continuous function:

$$h(t):=L(e^{it})-it$$

18.For every $t,$ one has $e^{h(t)}=1,$ so $h(t)$ belongs to the discrete set $2\pi i\mathbb{Z}.$ A continuous function from an interval into a discrete set is constant. Both endpoints of the path are the point $1,$ because $e^0=e^{2\pi i}=1,$ but:

$$h(0)=L(1),\qquad h(2\pi)=L(1)-2\pi i$$

19.The two values differ by $2\pi i,$ which contradicts constancy. Along one circuit of the unit circle, a continuous argument changes by $2\pi,$ although the starting and ending point are both $1.$

20.A branch cut is a deleted curve that prevents such a circuit. After a ray from the origin to infinity has been removed, the remaining plane has a continuous choice of argument. Fix an angle $\beta$ and remove the ray:

$$R_{\beta}:=\{\ re^{i\beta}\mid r\geq0\ \}$$

21.Every point of $\mathbb{C}\setminus R_{\beta}$ has a unique argument $\mathrm{Arg}_{\beta}(z)$ in the interval $(\beta,\beta+2\pi).$ The branch for this interval is:

$$L_{\beta}(z):=\ln|z|+i\mathrm{Arg}_{\beta}(z)$$

22.For $\beta=-\pi,$ the removed ray is the non-positive real axis and the selected angles are in $(-\pi,\pi).$ This branch is the principal branch on:

$$\mathbb{C}\setminus(-\infty,0]$$

23.The principal value $\mathrm{Log}(z)$ is defined pointwise on every $z\neq0,$ including negative real numbers. Its restriction to the cut plane is continuous and complex differentiable, so this restriction is the principal branch. On the whole punctured plane $\mathbb{C}^{*},$ the principal value is not a logarithmic branch.

24.For a fixed $x<0,$ the values on opposite sides of the cut have the limits:

$$ \begin{align} \lim_{y\to0^+}\mathrm{Log}(x+iy)&=\ln|x|+i\pi\\[6pt] \lim_{y\to0^-}\mathrm{Log}(x+iy)&=\ln|x|-i\pi \end{align} $$

25.The jump across the negative real axis is $2\pi i.$ With a different branch cut, the jump is on a different curve. No domain of a single-valued branch can contain a closed path that winds a nonzero number of times around the origin.

26.If $L_1$ and $L_2$ are two branches on the same connected domain, their difference has values in $2\pi i\mathbb{Z},$ because $e^{L_1(z)-L_2(z)}=1.$ The difference is continuous, and the domain is connected, so it is constant. This constant has the form $2m\pi i$ for some fixed integer $m:$

$$L_1(z)-L_2(z)=2m\pi i$$

Logarithmic identities and their restrictions

27.For the multivalued logarithm, the product law is an equality of sets. For nonzero $z_1$ and $z_2,$ one has:

$$\log(z_1z_2)=\log z_1+\log z_2$$

28.The right-hand side is the set of all sums of one logarithm of each factor. If $\theta_1$ and $\theta_2$ are arguments of $z_1$ and $z_2,$ these sums have the form:

$$\ln|z_1|+\ln|z_2|+i(\theta_1+\theta_2+2k\pi)$$

29.Since $|z_1z_2|=|z_1||z_2|$ and $\theta_1+\theta_2$ is an argument of $z_1z_2,$ these are exactly the logarithms of the product.

30.For principal values, the same formula can fail because the sum of two principal arguments can leave $(-\pi,\pi].$ The correct formula is:

$$\mathrm{Log}(z_1z_2)=\mathrm{Log}(z_1)+\mathrm{Log}(z_2)-2m\pi i$$

31.The integer $m$ is the unique member of $\{\ -1,0,1\ \}$ for which $\mathrm{Arg}(z_1)+\mathrm{Arg}(z_2)-2m\pi$ is in $(-\pi,\pi].$ The product law has no correction when:

$$-\pi<\mathrm{Arg}(z_1)+\mathrm{Arg}(z_2)\leq\pi$$

32.For example, $\mathrm{Log}(-1)=i\pi$ and $(-1)(-1)=1.$ The two sides are:

$$ \begin{align} \mathrm{Log}((-1)(-1))&=\mathrm{Log}(1)=0\\[6pt] \mathrm{Log}(-1)+\mathrm{Log}(-1)&=2\pi i \end{align} $$

33.The two expressions differ by $2\pi i.$ The quotient law and the formula for the principal logarithm of an integer power have analogous correction terms in $2\pi i\mathbb{Z}.$ Before applying a real logarithm law to principal complex logarithms, one must check whether the selected arguments cross the branch cut.

Derivative and local power series

34.Every logarithmic branch on an open domain is complex differentiable. Fix a point $z_0$ in the domain of $L.$ The derivative of the exponential at $L(z_0)$ is $e^{L(z_0)}=z_0\neq0.$ By the local inverse function theorem, the exponential is injective on some neighborhood $V$ of $L(z_0),$ and its inverse on the image of $V$ is complex differentiable. Since $L$ is continuous, a sufficiently small neighborhood $W$ of $z_0$ is contained in this image and has $L(W)\subseteq V.$ For each $z\in W,$ both $L(z)$ and the local inverse are in $V$ and have exponential $z,$ so they are equal. Hence $L$ is complex differentiable near every point of its domain.

35.Differentiating the identity $e^{L(z)}=z$ with the chain rule, we have:

$$ \begin{align} e^{L(z)}L'(z)&=1\\[6pt] zL'(z)&=1\\[6pt] L'(z)&=\frac{1}{z} \end{align} $$

36.The derivative is independent of the branch because two branches on a connected domain differ by a constant multiple of $2\pi i.$

37.Near $z=1,$ the principal branch has a power series. For $|u|<1,$ the geometric series for $1/(1+u)$ converges. Its termwise antiderivative with value $0$ at $u=0$ is $\mathrm{Log}(1+u):$

$$\mathrm{Log}(1+u)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{u^n}{n}=u-\frac{u^2}{2}+\frac{u^3}{3}-\frac{u^4}{4}+\cdots$$

38.The condition $|u|<1$ means that $1+u$ is in the open disk centered at $1$ with radius $1.$ This disk contains neither the origin nor any point of the branch cut, so the sum of the series is $\mathrm{Log}(1+u)$ throughout the disk.

Connection with complex powers

39.Once a logarithmic branch $L$ has been fixed on a domain $U,$ the corresponding power for $a\in\mathbb{C}$ and $z\in U$ is:

$$z^a:=e^{aL(z)}$$

40.If $L$ is replaced by the branch $L+2k\pi i,$ the value is multiplied by $e^{2k\pi ia}.$ When $a$ is an integer, this factor is $1,$ so the value is independent of the branch. For a general complex exponent, the power can have different values on different branches.

41.For $i^i,$ all logarithms of $i$ are:

$$\log i=\left\{\ i\left(\frac{\pi}{2}+2k\pi\right)\mid k\in\mathbb{Z}\ \right\}$$

42.The corresponding values of $i^i$ are:

$$i^i=\left\{\ e^{-\pi/2-2k\pi}\mid k\in\mathbb{Z}\ \right\}$$

43.Every value in this set is a positive real number. For the principal logarithm, $k=0,$ so the principal value of $i^i$ is $e^{-\pi/2}.$