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mathematics_basic_for_ML
  • README
  • README
    • Summary
    • Geometry
      • EulerAngle
      • Gimbal lock
      • Quaternion
      • RiemannianManifolds
      • RotationMatrix
      • SphericalHarmonics
    • Information
      • Divergence
      • 信息熵 entropy
    • LinearAlgebra
      • 2D仿射变换(2D Affine Transformation)
      • 2DTransformation
      • 3D变换(3D Transformation)
      • ComplexTransformation
      • Conjugate
      • Hessian
      • IllConditioning
      • 逆变换(Inverse transform)
      • SVD
      • det
      • eigendecomposition
      • 矩阵
      • norm
      • orthogonal
      • special_matrix
      • trace
      • vector
    • Mathematics
      • Complex
      • ExponentialDecay
      • average
      • calculus
      • convex
      • derivative
      • 距离
      • function
      • space
      • Formula
        • euler
        • jensen
        • taylor
        • trigonometric
    • Numbers
      • 几何级数
      • SpecialNumbers
    • NumericalComputation
      • ConstrainedOptimization
      • GradientDescent
      • Newton
      • Nominal
      • ODE_SDE
      • Preprocessing
    • Probability
      • bayes
      • distribution
      • expectation_variance
      • 贝叶斯公式
      • functions
      • likelihood
      • mixture_distribution
      • 一些术语
      • probability_distribution
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  1. README
  2. Mathematics
  3. Formula

euler

x为实数,有:

eix=cos⁡x+isin⁡xe^{ix} = \cos x + i\sin xeix=cosx+isinx

令$x = \pi$得:$e^{i\pi} + 1 = 0$ 令$x = \omega t$得,$e^{i\omega t} = \cos \omega t + i\sin\omega t$ 这个信号称为复指数信号,其实部为余弦信号,虚部为正弦信号。它可以理解为一个点在复平面上以角速度$\omega$逆时针运动。

cos⁡(nωt)=exp⁡(jωt)+exp⁡(−jωt)2sin⁡(nωt)=exp⁡(jωt)−exp⁡(−jωt)2\begin{aligned} \cos(n\omega t) = \frac{\exp(j\omega t) + \exp(-j\omega t)}{2} \\ \sin(n\omega t) = \frac{\exp(j\omega t) - \exp(-j\omega t)}{2} \end{aligned}cos(nωt)=2exp(jωt)+exp(−jωt)​sin(nωt)=2exp(jωt)−exp(−jωt)​​
PreviousFormulaNextjensen

Last updated 2 years ago

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