Encyclopedia
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 formula states that, for any real number
x,
-
where
- ' is the base of the natural logarithm
- ' is the imaginary unit
and are
trigonometric functions.
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 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 proof is based on the
Taylor series expansions of the
exponential function 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
radians — which is defined up to addition of 2p.
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:
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 , 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 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 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 2
nd order linear
ordinary differential equation is constructed:
-
or
-
Being a 2
nd order differential equation, there are two linearly independent solutions that satisfy it:
-
-
Both and 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 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
...
...
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
