Euler's Formula
Introduction
Euler's formula, named after the Swiss mathematician Euler, is a mathematical equation that establishes a deep relationship between trigonometric functions and the complex exponential function. Euler's formula states that for any real number x,
e^(ix) = cos(x) + i*sin(x)
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the cosine and sine functions, respectively. This formula can be derived by expanding the exponential function e^(ix) for x real, using the power series definition and the fact that i^2 = -1.
Historical Background
Euler's formula is named after Leonhard Euler, who introduced it in 1748. The formula was not widely accepted until it was proven by Lagrange in 1777. Euler's formula is a special case of the more general formula e^(z) = cos(z) + i*sin(z), where z is a complex number.
Mathematical Derivation
The mathematical derivation of Euler's formula begins with the power series expansion of the exponential function, sin(x), and cos(x). The power series expansion of the exponential function is given by:
e^x = 1 + x/1! + x^2/2! + x^3/3! + x^4/4! + ...
The power series expansion of sin(x) and cos(x) are given by:
sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ... cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...
Substituting x with ix in the power series expansion of the exponential function gives:
e^(ix) = 1 + ix - (ix)^2/2! + (ix)^3/3! - (ix)^4/4! + ...
Simplifying this expression using the fact that i^2 = -1 yields:
e^(ix) = 1 + ix - x^2/2! - ix^3/3! + x^4/4! + ...
Separating the real and imaginary parts of this expression gives:
e^(ix) = (1 - x^2/2! + x^4/4! - ...) + i*(x - x^3/3! + x^5/5! - ...)
Comparing this with the power series expansions of cos(x) and sin(x) gives Euler's formula:
e^(ix) = cos(x) + i*sin(x)
Applications
Euler's formula has many applications in different areas of mathematics, including calculus, differential equations, and complex analysis. It is also used in physics, engineering, and signal processing.
In calculus, Euler's formula is used to derive the identities for the sine and cosine of a sum of angles. These identities are useful in simplifying the process of integrating and differentiating trigonometric functions.
In differential equations, Euler's formula is used to solve homogeneous linear differential equations with constant coefficients. The solutions to these equations are often expressed in terms of exponential functions, and Euler's formula allows these solutions to be expressed in terms of sine and cosine functions.
In complex analysis, Euler's formula is used to define the complex exponential function, which is a key concept in the field. The complex exponential function is used in the definition of the complex logarithm and the complex power function, and it is also used in the proof of the fundamental theorem of algebra.
In physics and engineering, Euler's formula is used in the analysis of waveforms and signals. The formula allows sinusoidal waveforms to be represented in a form that is convenient for analysis and manipulation.
See Also
Complex Numbers, Trigonometry, Exponential Function, Calculus, Differential Equations, Complex Analysis, Signal Processing
