### Complex numbers: Complex functions

### Complex logarithm

Finally, we introduce the complex logarithm. As discussed in the beginning, we want to take over as many properties of real functions as possible. In particular, we want the rule of calculation \[\ln(z)=\ln({r\cdot \e^{\varphi\cdot\ii}})=\ln(r)+ \varphi\cdot\ii\] for \(z=r\cdot \e^{\varphi\cdot\ii}\) a complex number not equal to \(0\) in polar form, to remain valid.

But there is a snag: with \(\varphi\) we can add every integer multiple of \(2\pi\) without changing the value of \(z\). This means that the logarithm does not satisfy the definition of a *function*, which states that to no more than one value can be assigned to each value of the *function argument*. We write "function argument" to show that we mean the argument of a function, not the *argument* of a complex number. This can be remedied by requiring that we choose #\varphi\in\ivoc{-\pi}{\pi}#, meaning that only work with the *principal value* of the argument #z#. Hence, the following definition for the complex logarithm \(\ln(z)\).

The complex logarithm

If #z# is a complex number unequal to #0#, then the **complex logarithm** of #z# is \[ \begin{array}{rcl} \ln(z)&= &\ln(|z|) + \arg(z)\cdot\ii \tiny.\end {array}\]

Hence, we have \(\ln(r\cdot\e^{\varphi\cdot\ii})=\ln(r)+ \varphi \cdot\ii\pmod{2k\cdot\pi\cdot\ii}\).

The complex numbers #\ln(r)+ \left(\varphi +2k\cdot\pi\right)\cdot\ii# all have the same image under #\exp#: \[\exp\left(\ln(r)+ \left(\varphi +2k\cdot\pi\right)\cdot\ii\right)=\exp\left(\ln(r)+ \varphi \cdot\ii\right)=r\cdot\e^{\varphi\cdot\ii}\tiny.\]

One often talks about the logarithm #\ln(z)# of #z# as a complex number #w# with the property that #\e^w=z#. Just like the *argument* this is not a function. One then speaks of a *polyvalent function,* because they can take on multiple values in a single point. But this is a term we will avoid.

Here we choose to define a specific value by using the *principal value* #\arg(z)# of the *argument* of #z#. That value is called, just as with the *argument* (the argument of a complex number, not to be confused with the *argument* of a function), the **principal value of the** logarithm.

Sometimes in the mathematical literature you enounter #\rm Log# for the function we have indicated with #\log#, and #\log(z)# is used to indicate any or all complex solutions #w# of the equation #\e^w=z#.

The choice of #\log# as a function is similar to the one of higher power roots of real numbers: there are two numbers #x# with #x^2=3#, which can both be called a root of #3#. (In general, at higher powers, more than #2# roots can occur.) But with #\sqrt{3}# we indicate the positive solution to the equation. This follows from the choice of the domain #x\ge0# for function #x^2#. By comparison: here we have defined the domain for #\exp# as the set of all complex numbers #z# to #-\pi\lt\Im(z)\le\pi#.

The inverse of the exponential function

- The exponential function #\exp# is injective on the subset #\left\{z\in\mathbb{C}\mid -\pi\lt \Im(z)\le\pi\right\}# of complex numbers #z# with #\Im(z)\in\ivoc{-\pi}{\pi}#.
- The range of #\exp# on this domain is #\mathbb{C}\setminus\{0\}#, the set of all complex numbers unequal to #0#.
- The complex logarithm #\ln# is the inverse function of the exponential function with this domain. Hence, it has domain #\mathbb{C}\setminus\{0\}# and reach #\left\{z\in\mathbb{C}\mid -\pi\lt \Im(z)\le\pi\right\}#.

*Proof:* Assume that #z# and #w# are complex numbers with #\exp(z)=\exp(w)#. Then #\e^z=\e^w# applies, and because of the *rules of calculation for complex powers*, #\e^{zw}=1#. The definition *of complex powers of the number of Euler* shows that \[\eqs{ \e^{\Re(zw)}&=&\left|\e^{zw}\right|=\left| 1\right|=1\cr \Im(zw)&=&\arg\left(\e^{zw}\right)=0\pmod{2\pi}\cr}\] such that #zw=\Re(zw)+\Im(zw)\cdot\ii=0\pmod{2\pi\cdot\ii}#, or #z=w\pmod{2\pi\cdot\ii}#. If #z# and #w# both lie in the domain #\left\{z\in\mathbb{C}\mid -\pi\lt \Im(z)\le\pi\right\}#, then their imaginary parts will differ less than #2\pi# from each other, so that they must be equal. This proves #z=w#, and with it the injectivity of #\exp# as a function on the given domain.

It is known that each complex number unequal to #0# has a polar form and therefore can be written as #\e^z# for suitable #z# with #\Im(z)# in #\ivoc{-\pi}{\pi}#. This means that the image of #\exp# on the given domain is #\mathbb{C}\setminus\{0\}#.

The fact that #\ln# is the inverse of #\exp# on the given domain, follow from: \[ \begin{array}{rcl}\ln\left(\e^z\right)&=&\ln\left(\left|\e^{z}\right|\right) + \arg\left(\e^z\right)\cdot\ii\\ &&\phantom{uvwxyz}\color{blue}{\text{definition complex logarithm}}\\ &=&\ln\left(\e^{\Re(z)}\right) + \Im(z)\cdot\ii\\ &&\phantom{uvwxyz}\color{blue}{\text{definition complex function }\exp}\\&=&\Re(z) + \Im(z)\cdot\ii\\ &=&z\\ \end {array}\]

Rules of calculation for the complex logarithm

Let #z# and #w# two complex numbers unequal to #0# and let #n# be a natural number. Then holds:

- #\ln\left(z\cdot w\right)=\ln\left(z\right)+\ln\left(w\right)\pmod{2\pi\cdot\ii}#
- #\ln(z^n)=n\cdot \ln\left(z\right)\pmod{2\pi\cdot\ii}#
- #\e^{\ln\left(z\right)}=z#
- #\ln\left(\e^{z}\right)=z\pmod{2\pi\cdot\ii}#

The first equality follows from \[ \begin{array}{rcl}\ln\left(z\cdot w\right)&=& \ln\left(\left|z\cdot w\right|\right)+\arg\left(z\cdot w\right)\cdot\ii\\&=& \ln\left(\left|z\right|\right)+\ln\left(\left| w\right|\right)+\arg\left(z\right)\cdot\ii+\arg\left(w\right)\cdot\ii\pmod{2\pi\cdot\ii}\\&=&\ln(z)+\ln(w)\pmod{2\pi\cdot\ii}\end {array}\] The Second by repeated application of the first one with #z^k=z^{k-1}\cdot z# for #k=n,n-1,\ldots,2#.

The third follows from the fact that the logarithm is the inverse of #\exp#.

For #z# with #-\pi\lt \Im(z)\le\pi # the fourth equality also follows from the fact that the logarithm is the inverse of #\exp#. For other values of #z# the same conclusion cannot be draw, because #z# then not lies within the specially chosen domain of #\exp# where the function is injective. However, direct calculation shows that the complex logarithm behaves like the inverse, modulo #2\pi\cdot\ii#:

\[ \begin{array}{rcl}\ln\left(\e^{z}\right)&=& \ln\left(\left|\e^{z}\right|\right)+\arg\left(\e^{z}\right)\cdot\ii\\&&\phantom{uvwxyz}\color{blue}{\text{definition complex logarithm}}\\&=& \ln\left(\e^{\Re(z)}\right)+\left(\Im\left(z\right)\pmod{2\pi}\right)\cdot\ii\\&&\phantom{uvwxyz}\color{blue}{\text{definition complex function }\exp}\\&=&\left(\Re(z)+\Im(z)\cdot\ii\right) \pmod{2\pi\cdot\ii}\\&&\phantom{uvwxyz}\color{blue}{\text{definition real logarithm}}\\&=&z\pmod{2\pi\cdot\ii}\end {array}\]

This follows from the following calculation, in which the formula for #\arg(z)# from the theory

*real and imaginary part*is used.

\[ \begin{array}{rcl}\ln(-2.6-2.8\cdot\ii) &=& \ln\bigl(|-2.6-2.8\cdot\ii|\bigr) + \arg(-2.6-2.8\cdot\ii)\cdot\ii\\

&=& \ln\left(3.82\right) + 2\cdot \arctan\left(\frac{-2.8}{3.82-2.6}\right)\cdot\ii\pmod{2\pi\cdot\ii}\\

&&\phantom{xyz}\color{blue}{\sqrt{(-2.6)^2+(-2.8)^2}=3.82}\\ &=& 1.3 + 2\cdot \arctan\left(-2.30\right)\cdot\ii\\ &=& 1.3-2.3\cdot\ii\end {array}\]

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