Euler's Formula

Euler's Formula is a fundamental theorem in mathematics that relates the exponential function, trigonometric functions, and complex numbers. It was discovered by the Swiss mathematician Leonhard Euler in the 18th century and has since become a cornerstone of modern mathematics and science.

The Formula

The formula can be written as:

$e^{i\theta} = \cos(\theta) + i\sin(\theta)$

Here, $e$ is the mathematical constant e, $i$ is the imaginary unit, and $\theta$ is an angle in radians.

The formula relates two ways of representing complex numbers. The left-hand side represents a complex number in polar form, with the magnitude $e$ and the angle $\theta$. The right-hand side represents the same complex number in rectangular form, with the real part $\cos(\theta)$ and the imaginary part $i\sin(\theta)$.

Derivation

The derivation of Euler's Formula is not straightforward and requires some knowledge of calculus and complex numbers. Here, we provide a brief outline of the derivation.

Consider the power series expansions of the functions $e^{x}$, $\cos(x)$, and $\sin(x)$:

$e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}$

$\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n}}{(2n)!}$

$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n+1}}{(2n+1)!}$

Setting $x = i\theta$ in these series expansions, we obtain:

$e^{i\theta} = \sum_{n=0}^{\infty} \frac{(i\theta)^{n}}{n!}$

$\cos(\theta) = \sum_{n=0}^{\infty} \frac{(-1)^{n}\theta^{2n}}{(2n)!}$

$\sin(\theta) = \sum_{n=0}^{\infty} \frac{(-1)^{n}\theta^{2n+1}}{(2n+1)!}$

Next, we use the fact that $i^{2} = -1$ to simplify the expression for $e^{i\theta}$:

$e^{i\theta} = \sum_{n=0}^{\infty} \frac{(i\theta)^{n}}{n!} = \sum_{n=0}^{\infty} \frac{(i^{2}\theta^{2})^{n/2}}{(2n)!} + i\sum_{n=0}^{\infty} \frac{(i^{2}\theta^{2})^{(n-1)/2}}{(2n+1)!}$

$= \sum_{n=0}^{\infty} \frac{(-1)^{n}\theta^{2n}}{(2n)!} + i\sum_{n=0}^{\infty} \frac{(-1)^{n}\theta^{2n+1}}{(2n+1)!}$

Comparing this expression with the series expansions for $\cos(\theta)$ and $\sin(\theta)$, we obtain Euler's Formula:

$e^{i\theta} = \cos(\theta) + i\sin(\theta)$

Applications

Euler's Formula has many applications in mathematics and science. It is used in the study of complex analysis, the theory of waves, and the Fourier transform. It is also a key component of quantum mechanics and the study of electromagnetism.

In addition, Euler's Formula is often cited as one of the most beautiful equations in mathematics. It elegantly ties together seemingly unrelated mathematical concepts and has inspired generations of mathematicians and scientists.

Conclusion

Euler's Formula is a fundamental theorem in mathematics that relates the exponential function, trigonometric functions, and complex numbers. It is a key component of many areas of mathematics and science and is often cited as one of the most beautiful equations in mathematics. Its discovery by Leonhard Euler in the 18th century is a testament to the power and elegance of mathematical reasoning.

Links

オイラーの公式[JA]