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.)

  Part of a series of articles on
The mathematical constant, e
Natural logarithm
Applications in Compound interest Euler's identity & Euler's formula Half lives & Exponential growth/decay
Defining e Proof that e is irrational Representations of e Lindemann–Weierstrass theorem
People John Napier Leonhard Euler
Schanuel's conjecture


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



where

is the base of the natural logarithm


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 eix 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 ez (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 multi-valued function, 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 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 real-valued. 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 eix 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 ex, 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 by



This 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) by


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



because i 2 = −1 by definition. From this the following 2nd-order linear ordinary differential equation is constructed:

or


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



Both cos(x) and sin(x) are real functions in which the 2nd 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

References

1. ^ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley, p. 22-10. ISBN 0-201-02010-6. 
2. ^ John Stillwell (2002). Mathematics and Its History. Springer. 

External links

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.
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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).
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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.
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Leonhard Euler

Portrait by Johann Georg Brucker
Born March 15 1707(1707--)
Basel, Switzerland
Died September 18 [O.S.
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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".
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formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities.
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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.
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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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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).
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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) = ex 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
is Euler's number, the base of the natural logarithm,

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For , see .
The half-life of a quantity, subject to exponential decay, is the time required for the quantity to decay to half of its initial value.
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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

can be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function ey
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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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Leonhard Euler

Portrait by Johann Georg Brucker
Born March 15 1707(1707--)
Basel, Switzerland
Died September 18 [O.S.
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In mathematics, specifically transcendence theory, Schanuel's conjecture is the following statement:
Given any n complex numbers z1,...

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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) = ex 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(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 (1682-1716).
Born July 10, 1682
Burbage, Leicestershire
Died June 5, 1716
Cambridge, Cambridgeshire
Residence UK
Nationality British
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