Euler's formula states that, for any real number x, one has = ⁡ + ⁡, where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively.

欧拉公式 （英語： Euler's formula，又稱 尤拉公式）是 複分析 领域的公式，它将 三角函数 與 复指数函数 关联起来，因其提出者 莱昂哈德·歐拉 而得名。 歐拉公式提出，對任意 实数 ，都存在. 其中 是 自然对数的底数， 是 虚数單位，而 和 則是 餘弦 、 正弦 對應的 三角函数，参数 則以 弧度 为单位 [1]。 這一複數指數函數有時還寫作 cis x （英語： cosine plus i sine，余弦加 i 乘以正弦）。 由於該公式在 為 複數 時仍然成立，所以也有人將這一更通用的版本稱為歐拉公式 [2]。 …

Euler’s formula allows one to derive the non-trivial trigonometric identities quite simply from the properties of the exponential. For example, the addition for-mulas can be found as follows: cos( 1 + 2) =Re(ei( 1+ 2)) =Re(ei 1ei 2) =Re((cos 1 + isin 1)(cos 2 + isin 2)) =cos 1 cos 2 sin 1 sin 2 and sin( 1 + 2) =Im(ei( 1+ 2)) =Im(ei 1ei 2) =Im ...

Euler's Formula states that for any real x x, eix =cosx+isinx, e i x = cos x + i sin x, where i i is the imaginary unit, i = √−1 i = − 1. Euler's formula can be used to derive the following identities for the trigonometric functions sinx sin x and cosx cos x in terms of exponential functions:

在代数拓扑中， 欧拉示性数 （Euler characteristic）是一个 拓扑不变量 （事实上是 同伦不变量 ），对于一大类 拓扑空间 有定义，通常记作 二维拓扑 多面体 的欧拉示性数可以用以下公式计算：

Euler’s sine product first appears in §16 of his E41 (Enestrom Index) and the cotangent expansion in E61 (again¨ §16). Made rigorous over the complex numbers they are, respectively, applications of Weierstrass’s Factorisation Theorem and Mittag-Leffler’s Theorem (both 1876).

2024年2月9日 · Euler's Sine Identity can also be presented in the form: $\sin z = \dfrac 1 2 i \paren {e^{-i z} - e^{i z} }$ Also see. Euler's Cosine Identity; Euler's Tangent Identity; Euler's Cotangent Identity; Euler's Secant Identity; Euler's Cosecant Identity; Arcsine Logarithmic Formulation; Source of Name. This entry was named for Leonhard Paul Euler ...

Euler's formula allows for any complex number x x to be represented as e^ {ix} eix, which sits on a unit circle with real and imaginary components \cos {x} cosx and \sin {x} sinx, respectively. Various operations (such as finding the roots of unity) …

2025年3月14日 · for all x ∈R x ∈ R. for all z ∈ C z ∈ C. This entry was named for Leonhard Paul Euler. The Euler Formula for Sine Function was not put on an adequately rigorous footing until Karl Weierstrass provided a satisfactory proof for Weierstrass Factor Theorem.

2024年12月8日 · Leonhard Paul Euler famously published what is now known as Euler's Formula in $1748$. However, it needs to be noted that Roger Cotes first introduced it in $1714$, in the form: $\map \ln {\cos \theta + i \sin \theta} = i \theta$