higher order derivatives of sine and cosine

One may consider the sine and cosine either as real (http://planetmath.org/RealFunction) or complex functions.  In both cases they are everywhere smooth, having the derivatives of all orders (http://planetmath.org/OrderOfDerivative) in every point.  The formulae




where  n=0, 1, 2, (the derivative of the 0th order means the functionMathworldPlanetmath itself), can be proven by induction on n.  Another possibility is to utilize Euler’s formula, obtaining


here one has to compare the real (http://planetmath.org/ComplexFunction) and imaginary partsDlmfPlanetmath – supposing that x is real.

Title higher order derivatives of sine and cosine
Canonical name HigherOrderDerivativesOfSineAndCosine
Date of creation 2013-03-22 14:45:16
Last modified on 2013-03-22 14:45:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Derivation
Classification msc 26B05
Classification msc 46G05
Classification msc 26A24
Related topic FractionalDifferentiation
Related topic HigherOrderDerivatives
Related topic ExampleOfTaylorPolynomialsForSinX
Related topic CosineAtMultiplesOfStraightAngle