Euler's Formula

Euler's formula, named after Leonhard Euler , is a Mathematical Formula in Complex Analysis that shows a deep relationship between the Trigonometric Functions and the complex Exponential Function . ( Euler's Identity is a special case of the Euler formula.) Euler's formula states that, for any Real Number ''x'', : $e^{ix} = \cos(x) + i\sin(x) \!$ where : $e \,$ is the Base Of The Natural Logarithm : $i \,$ is the Imaginary Unit : $\mathrm{cos} \,$ and $\mathrm{sin} \,$ are Trigonometric Function s. Richard Feynman called Euler's formula "our jewel" and "the most remarkable formula in mathematics".¹ HISTORY Euler's formula was Proven for the first time by Roger Cotes in 1714 in the form : $\ln(\cos(x) + i\sin(x))=ix \$ (where "ln" means Natural Logarithm , i.e. log with base ''e'')². It was Euler who published the equation in its current form in arose only some 50 years later (see Caspar Wessel ). Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, '' Elements of Algebra '', he introduces these numbers almost at once and then uses them in a natural way throughout. APPLICATIONS IN COMPLEX NUMBER THEORY This formula can be interpreted as saying that the function ''e''^''ix'' traces out the Unit Circle in the Complex Number plane as ''x'' ranges through the real numbers. Here, ''x'' is the Angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in Radian s. The formula is valid only if sin and cos take their arguments in radians rather than in degrees. The original proof is based on the Taylor Series expansions of the Exponential Function ''e''^''z'' (where ''z'' is a complex number) and of sin ''x'' and cos ''x'' for real numbers ''x'' (see below). In fact, the same proof shows that Euler's formula is even valid for all ''complex'' numbers ''z''. Euler's formula can be used to represent complex numbers in Polar Coordinates . Any complex number ''z'' = ''x'' + ''iy'' can be written as
	:<math> \bar{z}	x - iy = z (\cos \phi - i\sin \phi ) = z e^{-i \phi} \,</math>
	:<math>z	\sqrt{x^2+y^2}</math> the Magnitude of ''z''
	Z	z e^{i \phi} =

e^{\ln z + i \phi}\,
: <math>\ln Z \ln z + i \phi\,</math>

& = rac{e^{i(x+y)}+e^{i(-x-y)}}{4}+rac{e^{i(x-y)}+e^{i(-x+y)}}{4} \
& = rac{\cos(x+y)}{2} + rac{\cos(x-y)}{2}.
\end{align}

Another technique is to represent the sinusoids in terms of the Real Part of a more complex expression, and perform the manipulations on the complex expression. For example:

:

\begin{align}
\cos(x\cdot n)+\cos(x(n-2)) & = \mathrm{Re} \{\quad e^{ix n}+e^{ix(n-2)}\quad \} \
& = \mathrm{Re} \{\quad e^{ix(n-1)}\cdot (e^{ix}+e^{-ix})\quad \} \
& = \mathrm{Re} \{\quad e^{ix(n-1)}\cdot 2\cos(x)\quad \} \
& = \cos(x(n-1))\cdot 2\cos(x).
\end{align}

OTHER APPLICATIONS

In Differential Equations , the function ''e''^''ix'' is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. Euler's Identity is an easy consequence of Euler's formula.

In Electrical Engineering and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions (see Fourier Analysis ), and these are more conveniently expressed as the real part of exponential functions with Imaginary exponents, using Euler's formula. Also, Phasor Analysis of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.

PROOFS

Using Taylor series

Here is a proof of Euler's formula using Taylor Series expansions
as well as basic facts about the powers of ''i'':

:

\begin{align}
i^0 &{}= 1, \quad &
i^1 &{}= i, \quad &
i^2 &{}= -1, \quad &
i^3 &{}= -i, \
i^4 &={} 1, \quad &
i^5 &={} i, \quad &
i^6 &{}= -1, \quad &
i^7 &{}= -i, \
\end{align}

and so on. The functions ''e''^''x'', cos(''x'') and sin(''x'') (assuming ''x'' is Real ) can be expressed using their Taylor expansions around zero:

:

\begin{align}
 e^x &{}= 1 + x + rac{x^2}{2!} + rac{x^3}{3!} + \cdots \
 \cos x &{}= 1 - rac{x^2}{2!} + rac{x^4}{4!} - rac{x^6}{6!} + \cdots \
 \sin x &{}= x - rac{x^3}{3!} + rac{x^5}{5!} - rac{x^7}{7!} + \cdots
\end{align}

For complex ''z'' we ''define'' each of these function by the above series, replacing ''x'' with ''z''. This is possible because the Radius Of Convergence of each series is infinite. We then find that

:

\begin{align}
 e^{iz} &{}= 1 + iz + rac{(iz)^2}{2!} + rac{(iz)^3}{3!} + rac{(iz)^4}{4!} + rac{(iz)^5}{5!} + rac{(iz)^6}{6!} + rac{(iz)^7}{7!} + rac{(iz)^8}{8!} + \cdots \
        &{}= 1 + iz - rac{z^2}{2!} - rac{iz^3}{3!} + rac{z^4}{4!} + rac{iz^5}{5!} - rac{z^6}{6!} - rac{iz^7}{7!} + rac{z^8}{8!} + \cdots \
        &{}= \left( 1 - rac{z^2}{2!} + rac{z^4}{4!} - rac{z^6}{6!} + rac{z^8}{8!} - \cdots 
ight) + i\left( z - rac{z^3}{3!} + rac{z^5}{5!} - rac{z^7}{7!} + \cdots 
ight) \
        &{}= \cos (z) + i\sin (z)
\end{align}

The rearrangement of terms is justified because each series is Absolutely Convergent . Taking ''z'' = ''x'' to be a real number gives the original identity as Euler discovered it.

Using calculus

Define the function

f

by

:

f(x) = rac{\cos x+i\sin x}{e^{ix}}. \

This is allowed since the equation

:

e^{ix}\cdot e^{-ix}=e^0=1 \

implies that

e^{ix}

is never zero.

The Derivative of

f

, according to the Quotient Rule , is:

:

\begin{align}
 f'(x) &{}= rac{(-\sin x+i\cos x)\cdot e^{ix} - (\cos x+i\sin x)\cdot i\cdot e^{ix}}{(e^{ix})^2} \
       &{}= rac{-\sin x\cdot e^{ix}-i^2\sin x\cdot e^{ix}}{(e^{ix})^2} \
       &{}= rac{(-1 - i^2) \cdot \sin x \cdot e^{ix}}{(e^{ix})^2} \
       &{}= rac{(-1 - (-1)) \cdot \sin x \cdot e^{ix}}{(e^{ix})^2} \
       &{}= 0.
\end{align}

Therefore,

f \

must be a Constant Function . Thus,

:

rac{\cos x + i \sin x}{e^{ix}}=f(x)=f(0)=rac{\cos 0 + i \sin 0}{e^0}=1.

Rearranging, it follows that

:

\displaystyle\cos x + i \sin x=e^{ix} .

Q.E.D.

Using ordinary differential equations

Define the function ''g''(''x'') by
:

g(x) \ \stackrel{\mathrm{def}}{=}\  e^{ix} .\

Considering that ''i'' is constant, the first and second derivatives of ''g''(''x'') are

:

g'(x) = i e^{ix} \

g''(x) = i^2 e^{ix} = -e^{ix} \

because ''i''² = −1 by definition. From this the following 2^nd-order Linear Ordinary Differential Equation is constructed:

:

g''(x) = -g(x) \

or
:

g''(x) + g(x) = 0. \

Being a 2^nd-order differential equation, there are two Linearly Independent solutions that satisfy it:

:

g_1(x) = \cos(x) \

g_2(x) = \sin(x). \

Both cos(''x'') and sin(''x'') are real functions in which the 2^nd derivative is identical to the negative of that function. Any Linear Combination of solutions to a Homogeneous differential equation is also a solution. Then, in general, the solution to the differential equation is

for any constants ''A'' and ''B''. But not all values of these two constants satisfy the known Initial Conditions for ''g''(''x''):

:

g(0) = e^{i0} = 1 \

g'(0) = i e^{i0} = i \

.

However these same initial conditions (applied to the general solution) are

:

g(0) = A \cos(0) + B \sin(0) = A \

g'(0) = -A \sin(0) + B \cos(0) = B \

resulting in

:

g(0) = A = 1 \

g'(0) = B = i \

and, finally,

:

g(x) \ \stackrel{\mathrm{def}}{=}\  e^{ix} = \cos(x) + i \sin(x). \

Q.E.D.

SEE ALSO

Leonhard Euler

Euler's Identity

Complex Number

Exponentiation

Exponential Function

Trigonometry

REFERENCES

EXTERNAL LINKS

Proof of Euler's Formula by Julius O. Smith III

Euler's Formula and Fermat's Last Theorem

Complex Exponential Function Module by John H. Mathews

Elements of Algebra

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Information About

Euler's Formula