Euler's formula
 This article is about Euler's formula in complex analysis. For Euler's formula in graph theory and polyhedral combinatorics see Euler characteristic. See also topics named after Euler.
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,
where
 is the imaginary unit
 and are trigonometric functions.
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(where "ln" means natural logarithm, i.e. log with base e)^{[2]}.
It was Euler who published the equation in its current form in 1748, basing his proof on the infinite series of both sides being equal. Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the complex plane 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 radians. 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
where
 the real part
 the imaginary part
 the magnitude of z
and is the argument of z— the angle between the x axis and the vector z measured counterclockwise and in radians — which is defined up to addition of 2π.
Now, taking this derived formula, we can use Euler's formula to define the logarithm 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 multivalued function, due to the fact that is multivalued.
Finally, the other exponential law
which can be seen to hold for all integers k, together with Euler's formula, implies several trigonometric identities as well as de Moivre's formula.
Relationship to trigonometry
Euler's formula provides a powerful connection between analysis and trigonometry, 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:
Complex exponentials can simplify trigonometry, because they are easier to manipulate than their sinusoidal components. One technique is simply to convert sinusoids into equivalent expressions in terms of exponentials. After the manipulations, the simplified result is still realvalued. For example:
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:
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: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:
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
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 byThis is allowed since the equation
implies that is never zero.
The derivative of , according to the quotient rule, is:
Therefore, must be a constant function. Thus,
Rearranging, it follows that
Q.E.D.
Using ordinary differential equations
Define the function g(x) byConsidering that i is constant, the first and second derivatives of g(x) are
because i^{ 2} = −1 by definition. From this the following 2^{nd}order linear ordinary differential equation is constructed:
Being a 2^{nd}order differential equation, there are two linearly independent solutions that satisfy it:
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):
 .
However these same initial conditions (applied to the general solution) are
resulting in
and, finally,
Q.E.D.See also
 Leonhard Euler
 Euler's identity
 Complex number
 de Moivre's formula
 Exponentiation
 Exponential function
 Trigonometry
References
1. ^ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. AddisonWesley, p. 2210. ISBN 0201020106.
2. ^ John Stillwell (2002). Mathematics and Its History. Springer.
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
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics.
..... Click the link for more information.In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological space's shape or structure. It is commonly denoted by (Greek letter chi).
..... Click the link for more information.Euler's formula, Euler's function, or Euler's identity. Euler's work touched upon so many fields that he is often the earliest written reference on a given matter.
..... Click the link for more information.Leonhard Euler
Portrait by Johann Georg Brucker
Born March 15 1707
Basel, Switzerland
Died September 18 [O.S.
..... Click the link for more information.Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
..... Click the link for more information.formula (plural: formulae, formulÃ¦ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities.
..... Click the link for more information.Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics.
..... Click the link for more information.trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
..... Click the link for more information.The exponential function is one of the most important functions in mathematics. The application of this function to a value x is written as exp(x).
..... Click the link for more information.Euler's identity, named after Leonhard Euler, is the equation
where is Euler's number, the base of the natural logarithm,
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e is the unique real number such that the value of the derivative (slope of the tangent line) of f(x) = e^{x} at the point x = 0 is exactly 1.
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The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.
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Compound interest is the concept of adding accumulated interest back to the principal, so that interest is earned on interest from that moment on. The act of declaring interest to be principal is called compounding (i.e. interest is compounded).
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Euler's identity, named after Leonhard Euler, is the equation
where
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where
 is Euler's number, the base of the natural logarithm,
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 For , see .
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In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the function's current size. Such growth is said to follow an exponential law (but see also Malthusian growth model).
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A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and λ is a positive number called the decay constant.
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In mathematics, the series representation of Euler's number e
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The mathematical constant e can be represented in a variety of ways as a real number. Since e is an irrational number (see proof that e is irrational), it cannot be represented as a fraction, but it can be represented as a continued fraction.
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John Napier
Painting of John Napier
Born 1550
Merchiston Tower, in
Edinburgh, Scotland
Died 4 April 1617
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Painting of John Napier
Born 1550
Merchiston Tower, in
Edinburgh, Scotland
Died 4 April 1617
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Leonhard Euler
Portrait by Johann Georg Brucker
Born March 15 1707
Basel, Switzerland
Died September 18 [O.S.
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Portrait by Johann Georg Brucker
Born March 15 1707
Basel, Switzerland
Died September 18 [O.S.
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In mathematics, specifically transcendence theory, Schanuel's conjecture is the following statement:
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 Given any n complex numbers z_{1},...
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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e is the unique real number such that the value of the derivative (slope of the tangent line) of f(x) = e^{x} at the point x = 0 is exactly 1.
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In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . Its precise definition is dependent upon the particular method of extension.
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trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
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Richard Phillips Feynman
Richard Feynman, dust jacket photo for
What Do You Care What Other People Think?
Born May 11 1918
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Richard Feynman, dust jacket photo for
What Do You Care What Other People Think?
Born May 11 1918
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In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. A proof is a logical argument, not an empirical one. That is, one must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics.
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Roger Cotes
Roger Cotes (16821716).
Born July 10, 1682
Burbage, Leicestershire
Died June 5, 1716
Cambridge, Cambridgeshire
Residence UK
Nationality British
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Roger Cotes (16821716).
Born July 10, 1682
Burbage, Leicestershire
Died June 5, 1716
Cambridge, Cambridgeshire
Residence UK
Nationality British
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1711 1712 1713  1714  1715 1716 1717
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1680s 1690s 1700s  1710s  1720s 1730s 1740s
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