Complex numbers: Complex functions
Complex exponents
We have seen that with complex numbers we can add, subtract, multiply, divide, and take the exponent to an integer power. We will now show how we define the #\e#-power and the sine and cosine function in complex numbers. This leads to examples of complex functions, meaning, functions that have a domain and a range that is composed of complex numbers. Our first goal is to define the power #a^z# with as exponent an arbitrary complex number #z# for every positive real number #a#. We start with the special base #a=\e#, the Euler's number.
Complex powers of the number of Euler
For every complex number #z# we define the complex number #\e^z# by
\[ \begin{cases}
|\e^z|& = \ \e^{{\Re} (z)}\\
\arg (\e^z) & =\ {\Im} (z)\pmod{2\pi}
\end {cases}
\] The function that adds #z# to the complex number #\e^z#, is also called the exponential function and is indicated with #\exp# such that #\exp(z)=\e^z#.
In this definition we use the #\e#-power which is known from the real numbers.
If #z# is a real number, then #|\e^z|=\e^z# and #\arg (\e^z)=0#, which coincides with the known, real #\e#-power. Therefore the definition of the #\e#-power is extended from #\mathbb{R}# to #\mathbb{C}#. In other words, the domain of the function #\exp# is #\mathbb{C}#.
Because the real #\e#-power is never equal to #0#, the complex #\e#-power will also never be equal to #0#: \[|\e^z |=\e^{{\Re}(z)} \neq 0\tiny.\] A number whose absolute value is not equal to #0#, therefore is not equal to #0# either.
Euler's Formula For every real number #\varphi# we have: \[\e^{\varphi\cdot\ii}=\cos(\varphi )+\sin(\varphi )\cdot\ii\tiny.\]
Consequently, we have a very compact notation for the complex number given by the polar coordinates #r# for the absolute value and #\varphi# for the argument: \[r\cdot \e^{\varphi\cdot\ii}\tiny.\] We call this expression polar form of the complex number.
Proof: The right hand side has absolute value #1# and argument #\varphi#. We check that the left hand side hast the same absolute value and the same argument modulo #{2\pi}#.
\[ \begin{array}{rcl}\left|\e^{\varphi\cdot\ii}\right|&=&\e^0=1\\ \arg\left(\e^{\varphi\cdot\ii}\right) &=& \Im\left(\varphi\cdot\ii\right)=\varphi\pmod{2\pi}\end {array}\]
The last remark follows from the fact that the complex number with absolute value #r# and argument #\varphi# from the theory of polar coordinates is known as #r\cdot\cos(\varphi)+r\cdot\sin(\varphi)\cdot\ii#, which is equivalent to #r\cdot\left(\cos(\varphi)+\sin(\varphi)\cdot\ii\right)#, and now can be rewritten as #r\cdot\e^{\varphi\cdot\ii}#.
With help of the absolute value #|z|# and the main value of the argument #\arg(z)#. the polar form of #z# can also be written as
\[z=|z|\cdot\e^{\arg(z)\cdot\ii}\tiny.\]
After the power of #\e# we can of course also handle other bases too.
Complex power of a positive real number
Let #r# be a positive real number, and #z# a complex number. Then we write
\[ r^z=\e^{\ln(r)\cdot z}\tiny.\]
Example: #2^{\ii} = \e^{\ln(2)\cdot\ii} = \cos(\ln(2))+\sin(\ln(2))\cdot\ii#.
If #z# is a real number, then this definition is consistent with the known:
\[ r^z=\left(\e^{\ln(r)}\right)^{z}=\e^{\ln(r)\cdot z}\]
Later, when we introduce the complex logarithm, we can use #\ln(r)# as in the formula above to define the complex power of a complex number #r#. The possibilities for the base #r# in the formula are thus no longer limited to a positive real number. That is in part already happened for #z=\frac{1}{2}# and #r=-1#: when defining the root of a negative number , we have introduced the root of #-1# as #\sqrt{-1}=\ii#, such that #(-1)^{\frac{1}{2}}=\ii#. We have noticed that #-\ii# has #-1# as its square. In general, in order to define #r^{\frac{1}{2}}#, we have to make a choice for the sign. This choice is made by the complex logarithm.
This identifies the complex power of every positive real number.
\[\e^{2.0+2.0\,\mathrm{i}}=\e^{2.0}\cdot \e^{2.0\,\mathrm{i}}=\e^{2.0}\cdot \left(\cos(2.0)+\sin(2.0)\cdot\ii\right)\approx -3.1+6.7\cdot\ii\]
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