Euler's Formula

Overview

**Euler's formula **

» exponent form of complex number

» reiθreiθ=r(cosθ+isinθ)=r(cosθ+isinθ)

→ ee is the base of natural logarithm

→ abstracted based on the properties of polar form r(cosθ+isinθ)r(cosθ+isinθ)
*The coordinate form, polar form, and exponent form help to establish a generic representation a+iba+ib 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=cosx+isinx

* Differentiate this :*

dy=(-sinx+icosx)dx

*substitute -1=i2 *

dy=(i2sinx+icosx)dx

*i is taken out of brackets *

dy=(isinx+cosx)idx

*substitute y=cosx+isinx *

dy=yidx

dyy=idx

*integrate this*

lny=ix

*change logarithm to its equivalent exponent form *

y=eix

we started with y=cosx+isinx and ended with y=eix. This proves,

eix=cosx+isinx

** Euler's Formula :**

eix=cosx+isinx

To develop intuitive understanding of Euler's formula, let us examine the properties of the form of complex number r(cosθ+isinθ).

Multiplication of two complex number leads to addition of θ -- similar to exponents where multiplication of two numbers leads to addition of powers

Prove that cosθ+isinθ has the properties of a number in exponent form * (Students are required to work this out) *

(cosθ1+isinθ1)×(cosθ2+isinθ2)

=(cos(θ1+θ2)+isin(θ1+θ2))

(cosθ+isinθ)n

=(cosnθ+isinnθ)

Similarly

eiθ1×eiθ2

=ei(θ1+θ2)

[eiθ]n

=einθ

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 deaxdx=aeax.

Prove that rate of change of z=cosθ+isinθ is iz.

dzdθ

=-sinθ+icosθ

=i2sinθ+icosθ

=i(isinθ+cosθ)

=iz

Similarly

ddθeiθ

=ieiθ

Intuitive understanding of Euler's formula :

On examining the properties of the form of complex number r(cosθ+isinθ).

• The representation is defined by ordered pair of numbers (r,θ)

• Multiplication of two complex number leads to addition of θ -- 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 deaxdx=aeax.

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 3√4, 4√4, 5√4, etc. Note that, 20+2√3-44√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+ib. Each of ii, 33+4i, etc. can be represented in the form a+ib. 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+bi is equivalently given as

=r(cos(θ+2nπ)

+isin(θ+2nπ))

=rei(θ+2nπ)

where r=√a2+b2,θ=tan-1(ba)

*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+bi Where b=0'.

summary

**Real number in Complex Number System : ** A real number a is equivalently given as

a+0i

=a(cos2nπ+isin2nπ)

=aei2nπ, where n=0,1,2,3...

Outline

The outline of material to learn "complex numbers" is as follows.

Note : * Click here for detailed overview of Complex-Numbers *

→ __Complex Numbers in Number System__

→ __Representation of Complex Number (incomplete)__

→ __Euler's Formula__

→ __Generic Form of Complex Numbers__

→ __Argand Plane & Polar form__

→ __Complex Number Arithmetic Applications__

→ __Understanding Complex Artithmetics__

→ __Addition & Subtraction__

→ __Multiplication, Conjugate, & Division__

→ __Exponents & Roots__

→ __Properties of Addition__

→ __Properties of Multiplication__

→ __Properties of Conjugate__

→ __Algebraic Identities__