Euler's Formula/Real Domain

Theorem

Let $\theta \in \R$ be a real number.

Then:

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

where:

$e^{i \theta}$ denotes the complex exponential function
$\cos \theta$ denotes the real cosine function
$\sin \theta$ denotes the real sine function
$i$ denotes the imaginary unit.

Corollary

$e^{-i \theta} = \cos \theta - i \sin \theta$

Proof 1

Consider the differential equation:

$D_z \map f z = i \cdot \map f z$

Step 1

We will prove that $z = \cos \theta + i \sin \theta$ is a solution.

 $\ds z$ $=$ $\ds \cos \theta + i \sin \theta$ $\ds \frac {\d z} {\d \theta}$ $=$ $\ds -\sin \theta + i \cos \theta$ Derivative of Sine Function, Derivative of Cosine Function, Linear Combination of Derivatives $\ds$ $=$ $\ds i^2 \sin \theta + i\cos \theta$ $i^2 = -1$ $\ds$ $=$ $\ds i \paren {i \sin \theta + \cos \theta}$ $\ds$ $=$ $\ds i z$

$\Box$

Step 2

We will prove that $y = e^{i\theta}$ is a solution.

 $\ds y$ $=$ $\ds e^{i\theta}$ $\ds \frac {\d y} {\d \theta}$ $=$ $\ds i e^{i \theta}$ Derivative of Exponential Function, Chain Rule for Derivatives, Linear Combination of Derivatives $\ds$ $=$ $\ds i y$

$\Box$

Step 3

Consider the initial condition $\map f 0 = 1$.

 $\ds \bigvalueat y {\theta \mathop = 0}$ $=$ $\ds e^{0 i}$ $\ds$ $=$ $\ds 1$ $\ds \bigvalueat z {\theta \mathop = 0}$ $=$ $\ds \cos 0 + i \sin 0$ $\ds$ $=$ $\ds 1$

So $y$ and $z$ are both particular solutions.

But a particular solution to a differential equation is unique.

Therefore $y = z$, that is, $e^{i \theta} = \cos \theta + i \sin \theta$.

$\blacksquare$

Proof 2

This:

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

is logically equivalent to this:

$\dfrac {\cos \theta + i \sin \theta} {e^{i \theta} } = 1$

for every $\theta$.

Note that the left expression is nowhere undefined.

Taking the derivative of this:

 $\ds \dfrac \d {\d \theta} e^{-i \theta} \paren {\cos \theta + i \sin \theta}$ $=$ $\ds e^{-i \theta} \paren {-\sin \theta + i \cos \theta} + \paren {-i e^{-i \theta} } \paren {\cos \theta + i \sin \theta}$ Product Rule for Derivatives and Derivative of Exponential Function $\ds$ $=$ $\ds e^{-i \theta} \paren {-\sin \theta + i \cos \theta - i \cos \theta - i^2 \sin \theta}$ $\ds$ $=$ $\ds e^{-i \theta} \paren {-\sin \theta + i \cos \theta - i \cos \theta + \sin \theta}$ $\ds$ $=$ $\ds e^{-i \theta} \paren 0$ $\ds$ $=$ $\ds 0$

Thus the expression, as a function of $\theta$, is constant and so yields the same value for every $\theta$.

We know the value at at least one point, that is, when $\theta = 0$:

$\dfrac {\cos 0 + i \sin 0} {e^{0 i}} = \dfrac {1 + 0} 1 = 1$

Thus it is $1$ for every $\theta$, which verifies the above.

Hence the result.

$\blacksquare$

Proof 3

It follows from Argument of Product equals Sum of Arguments that the $\map \arg z$ function for all $z \in \C$ satisfies the relationship:

$\map \arg {z_1 z_2} = \map \arg {z_1} + \map \arg {z_2}$

which means that $\map \arg z$ 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 $\map \arg z$ can not be considered a generalization to complex values of the ordinary $\log$ function for real values, since for $x \in \R$, we have:

$0 = \map \arg x \ne \log x$

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

$\map {\operatorname{alog} } z = a \map \arg z + \log \cmod z$

for any $a \in \C$, where $\cmod z$ 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:

$\map {\operatorname{alog} } z = a \map \arg z + \log \cmod z$

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

$\map {\operatorname{alog} } {z_1 z_2} = \map {\operatorname{alog} } {z_1} + \map {\operatorname{alog} } {z_2}$

and:

$\map {\operatorname{alog} } x = \log x$

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:

 $\ds \map {\operatorname{alog} } {z_1 z_2}$ $=$ $\ds a \map \arg {z_1 z_2} + \log \cmod {z_1 z_2}$ $\ds$ $=$ $\ds a \paren {\map \arg {z_1} + \map \arg {z_1} } + \map \log {\cmod {z_1} \cmod {z_2} }$ $\ds$ $=$ $\ds a \paren {\map \arg {z_1} + \map \arg {z_1} } + \log \cmod {z_1} + \log \cmod {z_2}$ $\ds$ $=$ $\ds a \map \arg {z_1} + \log \cmod {z_1} + a \map \arg {z_2} + \log \cmod {z_2}$ $\ds$ $=$ $\ds \map {\operatorname{alog} } {z_1} + \map {\operatorname{alog} } {z_1}$

Second part of our lemma is even more straightforward since for $x \in \R$, we have:

$\map \arg x = 0$

Then:

 $\ds \map {\operatorname{alog} } x$ $=$ $\ds a \map \arg x + \log \cmod x$ $\ds$ $=$ $\ds \log x$

which concludes the proof of Lemma 1.

$\Box$

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 $\log$:

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

Lemma 2

Let $\map {\operatorname{alog} } z = a \map \arg z + \log \cmod z$.

Then if:

$\dfrac {\map \d {\operatorname{alog} z} } {\d z} = \dfrac 1 z$

we must have:

$a = i$

Proof of Lemma 2

Let $z \in \C$ be such that:

$\cmod z = 1$

and:

$\map \arg z = \theta$

Then:

$z = \cos \theta + i \sin \theta$

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

 $\ds \map {\operatorname{alog} } z$ $=$ $\ds a \map \arg {\cos \theta + i \sin \theta} + \log \cmod z$ $\ds$ $=$ $\ds a \theta + \log 1 = a \theta$

We now have:

$a \theta = \map {\operatorname{alog} } {\cos \theta + i \sin \theta}$

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

 $\ds \map {\frac \d {\d \theta} } {a \theta}$ $=$ $\ds \map {\frac \d {\d \theta} } {\map {\operatorname{alog} } {\cos \theta + i \sin \theta} }$ $\ds a$ $=$ $\ds \dfrac {\map \d {\cos \theta + i \sin \theta} } {\d \theta} \dfrac {\map \d {\map {\operatorname{alog} } {\cos \theta + i \sin \theta} } } {\map \d {\cos \theta + i \sin \theta} }$ Chain Rule for Derivatives $\ds a$ $=$ $\ds \paren {-\sin \theta + i \cos \theta} \frac 1 {\cos \theta + i \sin \theta}$ by hypothesis: $\dfrac {\map \d {\operatorname{alog} z} } {\d z} = \dfrac 1 z$

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.

$\Box$

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

$\log z = i \map \arg z + \log \cmod z$

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

$\map \log {z_1 z_2} = \log z_1 + \log z_2$
$\log x = \log x$
$\dfrac {\map \d {\log z} } {\d z} = \dfrac 1 z$

Let its inverse function be referred to as the exponential of complex numbers, denoted as $e^z$.

If we write $z$ in its polar form:

$z = \cmod z \paren {\cos \theta + i \sin \theta}$

we have that:

$e^{i \theta + \log \cmod z} = \cmod z \paren {\cos \theta + i \sin \theta}$

Consider this equation for any number $z$ such that $\cmod z = 1$.

Then:

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

$\blacksquare$

Proof 4

Note that the following proof, as written, only holds for real $\theta$.

Define:

$\map x \theta = e^{i \theta}$
$\map y \theta = \cos \theta + i \sin \theta$

Consider first $\theta \ge 0$.

Taking Laplace transforms:

 $\ds \map {\laptrans {\map x \theta} } s$ $=$ $\ds \map {\laptrans {e^{i \theta} } } s$ $\ds$ $=$ $\ds \frac 1 {s - i}$ Laplace Transform of Exponential $\ds \map {\laptrans {\map y \theta} } s$ $=$ $\ds \map {\laptrans {\cos \theta + i \sin \theta} } s$ $\ds$ $=$ $\ds \map {\laptrans {\cos \theta} } s + i \, \map {\laptrans {\sin \theta} } s$ Linear Combination of Laplace Transforms $\ds$ $=$ $\ds \frac s {s^2 + 1} + \frac i {s^2 + 1}$ Laplace Transform of Cosine, Laplace Transform of Sine $\ds$ $=$ $\ds \frac {s + i} {\paren {s + i} \paren {s - i} }$ $\ds$ $=$ $\ds \frac 1 {s - i}$

So $x$ and $y$ have the same Laplace transform for $\theta \ge 0$.

Now define $\tau = -\theta, \sigma = -s$, and consider $\theta < 0$ so that $\tau > 0$.

Taking Laplace transforms of $\map x \tau$ and $\map y \tau$:

 $\ds \map {\laptrans {\map x \tau} } \sigma$ $=$ $\ds \frac 1 {\sigma - i}$ from above $\ds \map {\laptrans {\map y \tau} } \sigma$ $=$ $\ds \frac 1 {\sigma - i}$ from above

So $\map x \theta$ and $\map y \theta$ have the same Laplace transforms for $\theta < 0$.

The result follows from Injectivity of Laplace Transform.

$\blacksquare$

Proof 5

 $\ds \cos \theta + i \sin \theta$ $=$ $\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {\theta^{2 n} } {\paren {2 n}!} + i \sum_{n \mathop = 0}^\infty \paren {-1}^n \dfrac {\theta^{2 n + 1} } {\paren {2 n + 1}!}$ Definition of Complex Cosine Function and Definition of Complex Sine Function $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \paren {\paren {-1}^n \dfrac {\theta^{2 n} } {\paren {2 n}!} + i \paren {-1}^n \dfrac {\theta^{2 n + 1} } {\paren {2 n + 1}!} }$ Sum of Absolutely Convergent Series $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \paren {\dfrac {\paren {i \theta}^{2 n} } {\paren {2 n}!} + \dfrac {\paren {i \theta}^{2 n + 1} } {\paren {2 n + 1}!} }$ Definition of Imaginary Unit $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \dfrac {\paren {i \theta}^n} {n!}$ $\ds$ $=$ $\ds e^{i \theta}$ Definition of Complex Exponential Function

$\blacksquare$

Proof 6

 $\ds \int \frac {\d x} {\sqrt {x^2 + 1} }$ $=$ $\ds \map \ln {\sqrt {x^2 + 1} + x }$ Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form and $a = 1$ $\ds \int \frac {i\cos \theta \rd \theta } {\sqrt {1 - \sin^2 \theta} }$ $=$ $\ds \map \ln {\sqrt {1 - \sin^2 \theta} + i \sin \theta }$ $x = i \sin \theta$; $\d x = i\cos \theta \rd \theta$ $\ds \int \frac {i\cos \theta \rd \theta } {\sqrt {\cos^2 \theta} }$ $=$ $\ds \map \ln {\sqrt {\cos^2 \theta} + i \sin \theta }$ Sum of Squares of Sine and Cosine $\ds i\int \rd \theta$ $=$ $\ds \map \ln {\cos \theta + i \sin \theta }$ $\ds i \theta$ $=$ $\ds \map \ln {\cos \theta + i \sin \theta }$ $\ds e^{i \theta}$ $=$ $\ds \cos \theta + i \sin \theta$

$\blacksquare$

Examples

Example: $e^{i \pi / 4}$

$e^{i \pi / 4} = \dfrac {1 + i} {\sqrt 2}$

Example: $e^{i \pi / 2}$

$e^{i \pi / 2} = i$

Example: $e^{-i \pi / 2}$

$e^{-i \pi / 2} = -i$

Example: $e^{i \pi}$

$e^{i \pi} = -1$

Example: $e^{2 i \pi}$

$e^{2 i \pi} = 1$

Example: $e^{2 k i \pi}$

$\forall k \in \Z: e^{2 k i \pi} = 1$

Also known as

Euler's Formula in this and its corollary form are also found referred to as Euler's Identities, but this term is also used for the specific example:

$e^{i \pi} + 1 = 0$

It is wise when referring to it by name, therefore, to ensure that the equation itself is also specified.

Source of Name

This entry was named for Leonhard Paul Euler.