maths > complex-number

Euler's Formula

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Overview

Euler's formula

»  exponent form of complex number

»  reiθ$r {e}^{i \theta}$=r(cosθ+isinθ)$= r \left(\cos \theta + i \sin \theta\right)$
→  e$e$ is the base of natural logarithm
→  abstracted based on the properties of polar form r(cosθ+isinθ)$r \left(\cos \theta + i \sin \theta\right)$ The coordinate form, polar form, and exponent form help to establish a generic representation a+ib$a + i b$ for all complex numbers.

Euler's formula

At first, The Euler's Formula is proven in abstract form -- where manipulation of expressions and equations help to prove.

Then, the significance and the intuitive understanding of Euler's formula is discussed.

Proof for Euler's formula :
Consider
$y = \cos x + i \sin x$
Differentiate this :
$\mathrm{dy} = \left(- \sin x + i \cos x\right) \mathrm{dx}$
substitute $- 1 = {i}^{2}$
$\quad \mathrm{dy} = \left({i}^{2} \sin x + i \cos x\right) \mathrm{dx}$
$i$ is taken out of brackets
$\quad \mathrm{dy} = \left(i \sin x + \cos x\right) i \mathrm{dx}$
substitute $y = \cos x + i \sin x$
$\quad \mathrm{dy} = y i \mathrm{dx}$
$\quad \frac{\mathrm{dy}}{y} = i \mathrm{dx}$
integrate this
$\quad \ln y = i x$
change logarithm to its equivalent exponent form
$\quad y = {e}^{i x}$

we started with $y = \cos x + i \sin x$ and ended with $y = {e}^{i x}$. This proves,
${e}^{i x} = \cos x + i \sin x$

Euler's Formula :
${e}^{i x} = \cos x + i \sin x$

To develop intuitive understanding of Euler's formula, let us examine the properties of the form of complex number $r \left(\cos \theta + i \sin \theta\right)$.

Multiplication of two complex number leads to addition of $\theta$ -- similar to exponents where multiplication of two numbers leads to addition of powers

Prove that $\cos \theta + i \sin \theta$ has the properties of a number in exponent form (Students are required to work this out)

$\left(\cos {\theta}_{1} + i \sin {\theta}_{1}\right) \times \left(\cos {\theta}_{2} + i \sin {\theta}_{2}\right)$
$\quad \quad = \left(\cos \left({\theta}_{1} + {\theta}_{2}\right) + i \sin \left({\theta}_{1} + {\theta}_{2}\right)\right)$

${\left(\cos \theta + i \sin \theta\right)}^{n}$
$\quad \quad = \left(\cos n \theta + i \sin n \theta\right)$

Similarly
${e}^{i {\theta}_{1}} \times {e}^{i {\theta}_{2}}$
$\quad \quad = {e}^{i \left({\theta}_{1} + {\theta}_{2}\right)}$

${\left[{e}^{i \theta}\right]}^{n}$
$\quad \quad = {e}^{i n \theta}$

For an intuitive understanding of Euler's formula. Rate of change of the function with respect to theta equals $i$ times the function itself -- similar to base of natural logarithm where $\frac{d {e}^{a x}}{\mathrm{dx}} = a {e}^{a x}$.

Prove that rate of change of $z = \cos \theta + i \sin \theta$ is $i z$.
$\frac{\mathrm{dz}}{d \theta}$
$\quad \quad = - \sin \theta + i \cos \theta$
$\quad \quad = {i}^{2} \sin \theta + i \cos \theta$
$\quad \quad = i \left(i \sin \theta + \cos \theta\right)$
$\quad \quad = i z$

Similarly
$\frac{d}{d \theta} {e}^{i \theta}$
$\quad \quad = i {e}^{i \theta}$

Intuitive understanding of Euler's formula :

On examining the properties of the form of complex number $r \left(\cos \theta + i \sin \theta\right)$.

•  The representation is defined by ordered pair of numbers $\left(r , \theta\right)$

•  Multiplication of two complex number leads to addition of $\theta$ -- similar to exponents where multiplication of two numbers leads to addition of powers

•  rate of change with respect to theta equals $i$ times the function itself -- similar to base of natural logarithm where $\frac{d {e}^{a x}}{\mathrm{dx}} = a {e}^{a x}$.

These properties along with the abstract derivation given in a page earlier, it is understood that a complex number can equivalently be represented as an exponent with base $e$.

Significance of Euler's Formula:

Remember that irrational numbers have so many different variants like $\sqrt[3]{4}$, $\sqrt[4]{4}$, $\sqrt[5]{4}$, etc. Note that, $20 + \sqrt[2]{3} - 4 \sqrt[4]{3}$ is an irrational number.

Irrational numbers do not have an uniform representation and so they are represented with various forms of numerical expressions.

In case of complex numbers, all complex numbers can be represented in the form $a + i b$. Each of ${i}^{i}$, ${3}^{3 + 4 i}$, etc. can be represented in the form $a + i b$. This is made possible by Euler's Formula that connects the exponent form to the coordinate form.

This is explained in detail in the next page.

Polar form: A complex number in the form $a + b i$ is equivalently given as

$\quad \quad = r {e}^{i \left(\theta + 2 n \pi\right)}$
where $r = \sqrt{{a}^{2} + {b}^{2}} , \theta = {\tan}^{- 1} \left(\frac{b}{a}\right)$

The coordinate form of complex number is equivalently represented in an exponential form by Euler's Formula.

It is noted that the real numbers can also be represented in the form of $a + b i$ Where $b = 0$'.

summary

Real number in Complex Number System : A real number $a$ is equivalently given as
$a + 0 i$
$\quad \quad = a \left(\cos 2 n \pi + i \sin 2 n \pi\right)$
$\quad \quad = a {e}^{i 2 n \pi}$, where $n = 0 , 1 , 2 , 3. . .$

Outline