proof of addition formula of exp

The addition formulaPlanetmathPlanetmath


of the complex exponential function may be proven by applying Cauchy multiplication rule to the Taylor seriesMathworldPlanetmath expansions ( of the right side factors (  We present a proof which is based on the derivativePlanetmathPlanetmath of the exponential functionDlmfDlmfMathworldPlanetmathPlanetmath.

Let a be a complex constant.  Denote  ez=w(z).  Then  w(z)w(z).  Using the product ruleMathworldPlanetmath and the chain ruleMathworldPlanetmath we calculate:


Thus we see that the product  w(z1)w(a-z1)=ez1ea-z1  must be a constant A.  If we choose specially  z1=0,  we obtain:




If we denote  a-z1:=z2, the preceding equation reads  ez1ez2=ez1+z2. Q.E.D.

Title proof of addition formula of exp
Canonical name ProofOfAdditionFormulaOfExp
Date of creation 2013-03-22 16:32:03
Last modified on 2013-03-22 16:32:03
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 4
Author pahio (2872)
Entry type Proof
Classification msc 30D20
Related topic AdditionFormula
Related topic AdditionFormulas
Defines addition formula of exponential function