What is the history and significance of Euler's formula?

  • Thread starter Greg Bernhardt
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In summary, Euler's formula, ##e^{ix} = \cos x + i\sin x##, allows for the definition of trigonometric functions without relying on geometry. It can also be used to define cos and sin as ##\cos x = \frac{1}{2}(e^{ix} + e^{-ix})## and ##\sin x = \frac{1}{2i}(e^{ix} - e^{-ix})##, respectively. The formula was discovered by Cotes and has been attributed to Leonhard Euler. It is also known for its connection to the complex numbers and its elegant representation of the fundamental theorem of algebra.
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Definition/Summary

Euler's formula, [itex]e^{ix}\ =\ \cos x\ +\ i \sin x[/itex], enables the trigonometric functions to be defined without reference to geometry.

Equations

[tex]e^{ix}\ =\ \cos x\ +\ i \sin x[/tex]

and so cos and sin may be defined:

[tex]\cos x\ =\ \frac{1}{2}\left(e^{ix}\ +\ e^{-ix}\right)[/tex] and [tex]\sin x\ =\ \frac{1}{2i}\left(e^{ix}\ -\ e^{-ix}\right)[/tex]

or:

[tex]\cos x\ =\ 1\ -\ \frac{x^2}{2} +\ \frac{x^4}{24} -\ \frac{x^6}{720}\ \dots\ = \sum_{n\,=\,0}^{\infty}\frac{(-x)^{2n}}{(2n)!}[/tex]

[tex]\sin x\ =\ x\ -\ \frac{x^3}{6} +\ \frac{x^5}{120} -\ \frac{x^7}{5040}\ \dots\ = \sum_{n\,=\,0}^{\infty}\frac{(-x)^{2n+1}}{(2n+1)!}[/tex]

Extended explanation

Proof of Euler's formula, starting from the trignonometric definitions of cos and sin:

Using the chain rule:

[tex]\frac{d}{dx}\left(e^{-ix}\,(cosx\ +\ i sinx)\right)[/tex]

[tex]=\ e^{ix}\,(-i cosx\ +\ sinx\ -\ sinx\ +\ i cosx)[/tex]

[tex]=\ 0[/tex]

and so [itex]e^{-ix}\,(cosx\ +\ i sinx)[/itex] is a constant. Setting x = 0 we find that this constant must be 1.

and so [tex]cosx\ +\ i sinx\ =\ e^{ix}[/tex]

History:

Euler's formula was discovered by Cotes.

de Moivre's formula, [itex](cosx\ +\ i sinx)^n[/itex] = [itex]cos(nx)\ +\ i sin(nx)[/itex], is an obvious consequence of Euler's formula, but was discovered earlier.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
 
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Greg Bernhardt said:
Euler's formula was discovered by Cotes.
Wikipedia says something different:
Euler's formula appeared for the first time in 1748 in Leonhard Euler's two-volume introductio in analysin infinitorum, first under the premise that the angle is a real number. However, this limitation soon proved superfluous, because Euler's formula applies equally to all real and complex arguments.

Anyway, ##e^{i\pi}+1=0## is considered one of the most beautiful formulas in mathematics.

There are plenty of articles out there which deal with the subject, e.g. https://sites.math.washington.edu/~marshall/math_307/complexnos.pdf
 

Related to What is the history and significance of Euler's formula?

What is Euler's formula?

Euler's formula is a mathematical equation that relates the exponential function, trigonometric functions, and imaginary numbers. It is written as e^(ix) = cos(x) + isin(x), where e is the base of the natural logarithm, i is the imaginary unit, and x is any real number.

Who is Euler and why is this formula named after him?

Leonhard Euler was a Swiss mathematician who is known for his contributions to various areas of mathematics, including calculus, graph theory, and number theory. This formula is named after him because he first published it in his work "De Moivre's Theorem" in 1748.

What is the significance of Euler's formula?

Euler's formula is significant because it connects three important mathematical concepts - exponential, trigonometric, and imaginary numbers - in a single equation. It is also used in many areas of physics, engineering, and other sciences.

How is Euler's formula related to the unit circle?

Euler's formula is closely related to the unit circle, as it can be used to represent any point on the unit circle in the complex plane. The real part of the equation represents the x-coordinate, and the imaginary part represents the y-coordinate.

Can Euler's formula be extended to other types of functions?

Yes, Euler's formula can be extended to other types of functions, such as hyperbolic functions and power functions. This is known as the generalized Euler's formula, and it is written as e^(ix) = cos(x) + isin(x) + cosh(x) + isinh(x), where cosh is the hyperbolic cosine function and sinh is the hyperbolic sine function.

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