Euler's formula

Illustration on the unit circle in the complex number plane
Three-dimensional representation of Euler's formula

Euler's formula or Euler's formula , named after Leonhard Euler , or Euler's relation in some sources , is an equation that represents a fundamental connection between the trigonometric functions and the complex exponential functions using complex numbers .

Euler's formula

Euler's formula denotes the equation that is valid for everyone${\ displaystyle y \ in \ mathbb {R}}$

${\ displaystyle \ mathrm {e} ^ {\ mathrm {i} \, y} = \ cos \ left (y \ right) + \ mathrm {i} \, \ sin \ left (y \ right)}$,

where the constant , the Euler number (based on the natural exponential function or of the natural logarithm ), and the unit the imaginary unit denote the complex numbers. ${\ displaystyle \ mathrm {e}}$${\ displaystyle \ mathrm {i}}$

As a consequence of Euler's formula, the equation results for all of them${\ displaystyle z = {x + \ mathrm {i} y} \ in \ mathbb {C}}$

${\ displaystyle \ mathrm {e} ^ {z} = \ mathrm {e} ^ {x + \ mathrm {i} y} = \ mathrm {e} ^ {x} \ cdot \ mathrm {e} ^ {\ mathrm { i} y} = \ mathrm {e} ^ {x} \ cdot \ left (\ cos \ left (y \ right) + \ mathrm {i} \, \ sin \ left (y \ right) \ right)}$.

Derivation by means of series expansion

Animation of the derivation of Euler's formula

Euler's formula can be calculated from the Maclaurin series of functions and , , derive ${\ displaystyle \ mathrm {e} ^ {y}, \ sin y}$${\ displaystyle \ cos y}$${\ displaystyle y \ in \ mathbb {R}}$

{\ displaystyle {\ begin {aligned} \ mathrm {e} ^ {\ mathrm {i} y} & = 1+ \ mathrm {i} y + {(\ mathrm {i} y) ^ {2} \ over 2! } + {(\ mathrm {i} y) ^ {3} \ over 3!} + {(\ mathrm {i} y) ^ {4} \ over 4!} + \ dots \\ & = \ left (1 - {\ frac {y ^ {2}} {2!}} + {\ frac {y ^ {4}} {4!}} - \ dots \ right) + \ mathrm {i} \ cdot \ left (y - {\ frac {y ^ {3}} {3!}} + {\ frac {y ^ {5}} {5!}} - \ dots \ right) \\ & = \ cos (y) + \ mathrm {i} \ cdot \ sin (y) \ end {aligned}}}

The transformations are based on ${\ displaystyle i ^ {2} = - 1.}$

Euler's identity

Animation of the approximation of by the expression . The dots each for the values to represent.${\ displaystyle \ mathrm {e} ^ {\ mathrm {i} \ pi}}$${\ displaystyle \ lim _ {n \ rightarrow \ infty} (1+ \ mathrm {i} \ pi / n) ^ {n}}$${\ displaystyle n}$${\ displaystyle (1+ \ mathrm {i} \ pi / n) ^ {j}}$${\ displaystyle j = 0, \ dots, n}$

For , the so-called Euler's identity results from Euler's formula${\ displaystyle y = \ pi}$

${\ displaystyle \ mathrm {e} ^ {\ mathrm {i} \, \ pi} = {- 1}}$,

which establishes a simple connection between four of the most important mathematical constants : Euler's number , the circle number , the imaginary unit and the real unit . The following modified variant of the equation is sometimes preferred - although more complicated - because it adds another mathematically significant constant with zero : ${\ displaystyle \ mathrm {e}}$ ${\ displaystyle \ mathrm {\ pi}}$${\ displaystyle \ mathrm {i}}$${\ displaystyle 1}$

${\ displaystyle \ mathrm {e} ^ {\ mathrm {i} \, \ pi} + 1 = 0}$.

If one extends the definition of the numerical value as a limit value to the complex numerical level , the result is correspondingly for the value . The animation on the right shows the intermediate results of the calculation of the expression , which are connected to a segment in the complex plane : It shows that this segment assumes the shape of an arc, the left end of which actually approaches the number on the real axis. ${\ displaystyle \ mathrm {e} ^ {z}}$${\ displaystyle \ textstyle \ lim _ {n \ rightarrow \ infty} (1 + z / n) ^ {n}}$${\ displaystyle z \ in \ mathbb {C}}$${\ displaystyle z = \ mathrm {i} \ pi}$${\ displaystyle {-1}}$${\ displaystyle (1+ \ mathrm {i} \ pi / n) ^ {n}}$${\ displaystyle n}$${\ displaystyle {-1}}$

Relationship between exponentials and trigonometric functions

Relationship between sine, cosine and exponential function

formulation

Euler's formula is a central link between analysis and trigonometry :

${\ displaystyle \ sin x = {\ frac {\ mathrm {e} ^ {\ mathrm {i} x} - \ mathrm {e} ^ {- \ mathrm {i} x}} {2 \ mathrm {i}} }, \ quad \ cos x = {\ frac {\ mathrm {e} ^ {\ mathrm {i} x} + \ mathrm {e} ^ {- \ mathrm {i} x}} {2}}}$.

Derivation

Sine and cosine result from the real part and the imaginary part of the complex exponential function.

The real part is obtained by adding a complex number to the conjugate and dividing by two: ${\ displaystyle z}$ ${\ displaystyle {\ bar {z}}}$

${\ displaystyle \ cos (x) = \ mathrm {Re} (\ mathrm {e} ^ {\ mathrm {i} x}) = {\ frac {\ mathrm {e} ^ {\ mathrm {i} x} + \ mathrm {e} ^ {- \ mathrm {i} x}} {2}}}$.

The imaginary part is obtained by calculating: ${\ displaystyle {\ frac {z - {\ bar {z}}} {2 \ mathrm {i}}}}$

${\ displaystyle \ sin (x) = \ mathrm {Im} (\ mathrm {e} ^ {\ mathrm {i} x}) = {\ frac {\ mathrm {e} ^ {\ mathrm {i} x} - \ mathrm {e} ^ {- \ mathrm {i} x}} {2 \ mathrm {i}}}}$.

Explanation

The Euler formula allows a completely new view of the trigonometric functions, since the functions sine and cosine , which are used in conventional trigonometry only with real arguments, now also have a meaning in complex analysis.

The formulas for the real and imaginary part result from:

{\ displaystyle {\ begin {aligned} \ mathrm {Re} (a ​​+ b \, \ mathrm {i}) = {\ frac {z + {\ bar {z}}} {2}} = {\ frac {( a + b \ mathrm {i}) + (ab \, \ mathrm {i})} {2}} = {\ frac {2a} {2}} = a, \\\ mathrm {Im} (a + b \, \ mathrm {i}) = {\ frac {z - {\ bar {z}}} {2 \ mathrm {i}}} = {\ frac {(a + b \, \ mathrm {i}) - (from \, \ mathrm {i})} {2 \ mathrm {i}}} = {\ frac {2b \ mathrm {i}} {2 \ mathrm {i}}} = b \ end {aligned}}}

A consequence of the connection of trigonometric functions and exponential functions from the Euler formula is Moivresche's theorem (1730).

Hyperbolic functions

Providing the sines and cosines with imaginary arguments, a bridge to the hyperbolic functions is built :

${\ displaystyle \ sin (\ mathrm {i} y) = {\ mathrm {e} ^ {- y} - \ mathrm {e} ^ {y} \ over 2 \ mathrm {i}} = \ mathrm {i} \, {\ frac {\ mathrm {e} ^ {y} - \ mathrm {e} ^ {- y}} {2}} = \ mathrm {i} \, \ sinh (y)}$
${\ displaystyle \ cos (\ mathrm {i} y) = {\ frac {\ mathrm {e} ^ {- y} + \ mathrm {e} ^ {y}} {2}} = {\ frac {\ mathrm {e} ^ {y} + \ mathrm {e} ^ {- y}} {2}} = \ cosh (y)}$

As can be seen, the two functions obtained correspond exactly to the definitions of the hyperbolic sine and hyperbolic cosine .

Other uses

Vector representation of an alternating voltage in the complex plane

Based on this, Euler's formula can also be used to solve numerous other problems, for example when calculating the power of the imaginary unit with itself. Although the result obtained is ambiguous, all individual solutions remain in the real area with a main value of${\ displaystyle \ mathrm {i} ^ {\ mathrm {i}}}$${\ displaystyle \ mathrm {i} ^ {\ mathrm {i}} = \ mathrm {e} ^ {- \ pi / 2} = 0 {,} 207 \, 879 \ dots}$

A practically important application of Euler's formula can be found in the field of alternating current technology , namely in the investigation and calculation of alternating current circuits with the help of complex numbers.

history

Euler's formula first appeared in Leonhard Euler's two-volume Introductio in analysin infinitorum in 1748 under the premise that the angle is a real number. However, this restriction soon turned out to be superfluous, because Euler's formula applies equally to all real and complex arguments. This results from Euler's formula with a real argument in connection with the identity theorem for holomorphic functions .

Before that, Roger Cotes published an incorrect mathematical connection in 1714, which is similar to Euler's formula.

In modern notation it looks like this:

${\ displaystyle \ mathrm {i} \ cdot r \ cdot \ ln (\ cos (\ varphi) + \ mathrm {i} \ sin (\ varphi)) = r \ cdot \ varphi \ quad {\ text {(sic! )}}}$,

where a circle with a radius fixed in the coordinate origin and an angle between the x-axis and a ray that intersects the origin are considered. ${\ displaystyle r}$${\ displaystyle \ varphi}$

The imaginary unit should be on the other side of the equation. ${\ displaystyle \ mathrm {i}}$