Euler's Formula/Real Domain/Proof 3

Theorem

 * $e^{i \theta} = \cos \theta + i \sin \theta$

where $e^{i \theta}$ is the complex exponential function, $\cos$ is cosine, $\sin$ is sine, and $i$ is the imaginary unit.

Proof
It follows from Argument of Product is Sum of Arguments that the $\arg \left({z}\right)$ function for all $z \in \C$ satisfies the relationship:

which means that $\arg \left({z}\right)$ is a kind of logarithm, in the sense that it satisfies the fundamental property of logarithms: $\log x y = \log x + \log y$.

Notice that $\arg \left({z}\right)$ can not be considered a generalization to complex values of the ordinary $\log$ function for real values, since for $x \in \R$, we have $0 = \arg \left({x}\right) \ne \log x$.

If we do wish to generalize the $\log$ function to complex values, we can use $\arg \left({z}\right)$ to define a set of functions:


 * $\operatorname{alog} \left({z}\right) = a \arg \left({z}\right) + \log \left|{z}\right|$

for any $a \in \C$, where $|z|$ is the modulus of $z$.

All functions satisfy the fundamental property of logarithms and also coincide with the $\log$ function for all real values.

This is established in the following lemma.

Lemma 1
For all $a,z \in \C$, define the (complex valued) function $\operatorname{alog}$ as:


 * $\operatorname{alog} \left({z}\right) = a \arg \left({z}\right) + \log \left|{z}\right|$

then, for any $z_1, z_2 \in \C$ and $x \in \R$:


 * $\operatorname{alog} \left({z_1z_2}\right) = \operatorname{alog} \left({z_1}\right) + \operatorname{alog} \left({z_2}\right)$

and:


 * $\operatorname{alog} \left({x}\right) = \log x$

This means that our (complex valued) $\operatorname{alog}$ functions can genuinely be considered generalizations of the (real valued) $\log$ function.

Proof of Lemma 1
Let $z_1, z_2$ be any two complex numbers, straightforward substitution on the definition of $\operatorname{alog}$ yields:

Second part of our lemma is even more straightforward since for $x \in \R$, we have:
 * $\arg \left({x}\right) = 0$

Then:

which concludes the proof of Lemma 1.

We're left with an infinitude of possible generalizations of the $\log$ function, namely one for each choice of $a$ in our definition of $\operatorname{alog}$.

The following lemma proves that there's a value for $a$ that guarantees our definition of $\operatorname{alog}$ satisfies the much desirable property of $\log$:


 * $\dfrac{\mathrm d {\log x }}{\mathrm d x} = \dfrac 1 x$

Lemma 2
Let $\operatorname{alog} \left({z}\right) = a \arg \left({z}\right) + \log \left|{z}\right|$.

Then if:
 * $\dfrac {\mathrm d \left({\operatorname{alog} z}\right)} {\mathrm d z} = \dfrac 1 z$

we must have:
 * $a = i$

Proof of Lemma 2
Let $z \in \C$ be such that:
 * $\left|{z}\right| = 1$

and:
 * $\arg \left({z}\right) = \theta$

Then:
 * $z = \cos \theta + i \sin \theta$

Plugging those values in our definition of $\operatorname{alog}$:

We now have:


 * $a \theta = \operatorname{alog} \left({\cos \theta + i \sin \theta}\right)$

Taking the derivative with respect to $\theta$ on both sides, we have

This last equation is true regardless of the value of $\theta$.

In particular, for $\theta = 0$, we must have:
 * $a = i$

which proves the lemma.

We now have established there is one function which truly deserves to be called the logarithm of complex numbers, defined as:


 * $\log \left({z}\right) = i \arg \left({z}\right) + \log \left|{z}\right| $

Since for any $z,z_1,z_2\in\C,x\in\R$ it satisfies:


 * $\log (z_1 z_2) = \log (z_1) + \log (z_2)$
 * $\log (x) = \log{x}$
 * $\dfrac {\mathrm d \left({\log \left({z}\right)}\right)} {\mathrm d z} = \dfrac 1 z$

Lets call its inverse function the exponential of complex numbers, denoted as $e^z$.

If we write $z$ in its polar form:
 * $z = \left|{z}\right| \left({\cos \theta + i \sin \theta}\right)$

we have that:
 * $e^{i \theta + \log \left|{z}\right|} = \left|{z}\right| \left({\cos \theta + i \sin \theta}\right)$

Consider this equation for any number $z$ such that $\left|{z}\right| = 1$.

Then:


 * $e^{i \theta} = \cos \theta + i \sin \theta$