See Also

Euler's formula

Euler's formula, named after Leonhard Euler Leonhard Euler

Leonhard Euler was a Swiss [i] mathematician [i] and physicist [i]. ... 

, is a mathematical Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

 formula in complex analysis that shows a deep relationship between the trigonometric functions Trigonometric function

In mathematics [i], the trigonometric functions are function [i]s of an angle [i]; they are im ... 

 and the complex exponential function Exponential function

The exponential function is one of the most important function [i]s in mathematics [i]. ... 

. Euler's formula states that, for any real number x, where and are trigonometric function Trigonometric function

In mathematics [i], the trigonometric functions are function [i]s of an angle [i]; they are im ... 

s.

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Encyclopedia


Euler's formula, named after Leonhard Euler Leonhard Euler

Leonhard Euler was a Swiss [i] mathematician [i] and physicist [i]. ... 

, is a mathematical Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

 formula in complex analysis that shows a deep relationship between the trigonometric functions Trigonometric function

In mathematics [i], the trigonometric functions are function [i]s of an angle [i]; they are im ... 

 and the complex exponential function Exponential function

The exponential function is one of the most important function [i]s in mathematics [i]. ... 

.

Euler's formula states that, for any real number x,




where

' is the base of the natural logarithm E

The letter E is the fifth letter in the Latin alphabet [i]. ... 




' is the imaginary unit


and are trigonometric function Trigonometric function

In mathematics [i], the trigonometric functions are function [i]s of an angle [i]; they are im ... 

s.

History

Euler's formula was proven  for the first time by Roger Cotes in 1714, then rediscovered and popularized by Euler in 1748. Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the plane arose only some 50 years later .

Applications in complex number theory

This formula can be interpreted as saying that the function eix traces out the unit circle Unit circle

In mathematics [i], a unit circle is a circle [i] with unit [i] radius [i], i.e., a circle whose radiu ... 

 in the complex number Complex number

In mathematics [i], a complex number is a number [i] of the form
... 

 plane as x ranges through the real numbers. Here, x is the angle Angle

An angle is the figure formed by two rays [i] sharing a common endpoint [i], called the vertex [i]... 

 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 Radian

The radian is a unit of plane angle [i]. ... 

s. The formula is valid only if sin and cos take their arguments in radians rather than in degrees.

The proof is based on the Taylor series Taylor series

In mathematics [i], the Taylor series of an infinite [i]ly differentiable [i] real [i] ... 

 expansions of the exponential function Exponential function

The exponential function is one of the most important function [i]s in mathematics [i]. ... 

 ez and of sin x and cos x for real numbers x . In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.

Euler's formula can be used to represent complex numbers in polar coordinates. Any complex number z=x+iy can be written as

where
is the magnitude of z

and is the argument of z— the angle between the x axis and the vector z measured counterclockwise and in radian Radian

The radian is a unit of plane angle [i]. ... 

s — which is defined up to addition of 2p.

Now, taking this derived formula, we can use Euler's formula to define the logarithm Logarithm

The logarithm is the mathematical [i] operation that is the inverse [i] of ... 

 of a complex number. To do this, we also use the facts that
and
both valid for any complex numbers a and b.

Therefore, one can write:

for any . Taking the logarithm of both sides shows that:


and in fact this can be used as the definition for the complex logarithm. The logarithm of a complex number is thus a multi-valued function Multivalued function

In mathematics [i], a multivalued function is a total relation [i]; i.e. ... 

, due to the fact that is multi-valued.

Finally, the other exponential law


which can be seen to hold for all integers k, together with Euler's formula, implies several trigonometric identities List of trigonometric identities

In mathematics [i], trigonometric identities are equalities involving trigonometric function [i]s that a ... 

 as well as de Moivre's formula.

Relationship to trigonometry

Euler's formula provides a powerful connection between analysis and trigonometry Trigonometry

Trigonometry is a branch of mathematics [i] dealing with angle [i]s, triangle [i]s and trigonometric function [i] ... 

, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function:




The two equations above can be derived by adding or subtracting Euler's formulas:


and solving for either cosine or sine.

These formulas can even serve as the definition of the trigonometric functions for complex arguments x. For example, letting x = iy, we have:

Other applications

In differential equations Differential equation

In mathematics [i], a differential equation is an equation [i] in which the derivative [i]s of a function [i]... 

, the function eix is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. Euler's identity Euler's identity

In mathematical analysis [i], Euler's identity, named after Leonhard Euler [i], is the equation
... 

 is an easy consequence of Euler's formula.

In electrical engineering Electrical engineering

Electrical engineering is a professional engineering [i] discipline that deals with the study and appli ... 

 and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions , and these are more conveniently expressed as the real part of exponential functions with imaginary exponents, using Euler's formula.

Proofs


Using Taylor series

Here is a proof of Euler's formula using Taylor series Taylor series

In mathematics [i], the Taylor series of an infinite [i]ly differentiable [i] real [i] ... 

 expansions
as well as basic facts about the powers of i:




and so on. The functions ex, cos and sin can be written as:










and for complex z we define each of these function by the above series, replacing x with iz. This is possible because the radius of convergence of each series is infinite. We then find that













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.

Q.E.D.

Using calculus

Define the function by


This is allowed since the equation


implies that is never zero.

The derivative Derivative

In mathematics [i], the derivative is defined as the instantaneous rate of change of a function [i] ... 

 of is, according to the quotient rule:

  
  
  
  
  
  



Therefore, must be a constant function. Thus,

 
 
 



Q.E.D.

Using ordinary differential equations

Define the function by



Considering that is constant, the first and second derivatives of are




because by definition. From this the following 2nd order linear ordinary differential equation Ordinary differential equation

In mathematics [i], and particularly in analysis [i], an ordinary differential equati ... 

 is constructed:



or



Being a 2nd order differential equation, there are two linearly independent solutions that satisfy it:




Both and are real functions in which the 2nd derivative is identical to the negative of that function. Any linear combination of solutions to a homogeneous Ordinary differential equation

In mathematics [i], and particularly in analysis [i], an ordinary differential equati ... 

 differential equation is also a solution. Then, in general, the solution to the differential equation is

  
  



for any constants and But not all values of these two constants satisfy the known initial conditions for :

.


However these same initial conditions are




resulting in




and, finally,




Q.E.D.

See also

  • Leonhard Euler Leonhard Euler

    Leonhard Euler was a Swiss [i] mathematician [i] and physicist [i]. ... 

  • Euler's identity Euler's identity

    In mathematical analysis [i], Euler's identity, named after Leonhard Euler [i], is the equation

... 


  • Complex number Complex number

    In mathematics [i], a complex number is a number [i] of the form

... 


  • Exponential function Exponential function

    The exponential function is one of the most important function [i]s in mathematics [i]. ... 

  • Trigonometry Trigonometry

    Trigonometry is a branch of mathematics [i] dealing with angle [i]s, triangle [i]s and trigonometric function [i] ... 



References

  • Feynman, Richard P., The Feynman Lectures on Physics, vol. I Addison-Wesley , ISBN 0-201-02010-6, ISBN 02010211161

External links

  • by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
  • by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
  • by Craig Lewis.
  • by Julius O. Smith III