2.3 Complex e-function

Function:f(x)=ex,xR, e=2.7181

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We define the exponential function as (the only non trivial function) which is it’s own derivative:
Derivative of the exponential function:

dexdx=ex

Just using the definition of the factorial function we find the Taylor series expansion of the exponential function

ex=limn(1+xn)n=n=01n!xnEuler’s number: e

First we will prove some very important properties of the exponential function.
Fundamental addition formula of the exponential function:
Applying the definition of the e-function as a series we find exey=n=0xnn!m=0ymm!=n=0k=0nxkynkk!(nk)!=n=01n!k=0n(nk)xkynk=n=0(x+y)nn!=ex+y

Euler’s formula:
taking into account i4k=1,i4k+1=i,i4k+2=1,i4k+3=i, and using the definitions by the Taylor series we find eiφ=n=0(iφ)nn!=1+iφ+(iφ)22!+(iφ)33!+(iφ)44!+(iφ)55!+=1φ22!+φ44!++i(φφ33!+φ55!):=cosφ+isinφ

This is a very important relation. It can be understood as the definition of the sin and cos function and allows to replace cosφ and sinφ by the (complex) e-function (and vice versa) Simplification!!! (e.g. Waves =^complex e-function). In addition the symmetries of the sin and cos functions get already obvious.

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sinx and cosx vs. exp:
For real numbers cosx and sinx are just the symmetric resp. antisymmetric representation of the expx function with the following properties eix=cosx+isinxxReix=cosxisinxcosx=12(eix+eix)sinx=12i(eixeix)

Additionally we directly get

cos2x+sin2x=(cosx+isinx)(cosxisinx)=eixeix=1

Addition theorems for sin and cos functions:
Combining the exp-addition formula with Euler’s formula we find (cosycoszsinysinz)+i(cosysinz+sinycosz)=(cosy+isiny)(cosz+isinz)=eiyeiz=ei(y+z)=cos(y+z)+isin(y+z)

From real and imaginary part we finally get (representing the even and odd part of the complex exponential function) cosycoszsinysinz=cos(y+z)cosysinz+sinycosz=sin(y+z)

Combining both equations we easily get

tan(y+z)=tan(y)+tan(z)1tan(y)tan(z)

Back to complex numbers:


In general:
z=r(cosφ+isinφ)Re{z}=rcosφIm{z}=rsinφr=|z|

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Multiplication of complex numbers: z1z2=r1eiφ1r2eiφ2=(r1r2)ei(φ1+φ2)=(r1r2)(cos(φ1+φ2)+isin(φ1+φ2))zz¯=reiφreiφ=r2

Definition 6

f(z)=ezzC is the complex e-function with
z=a+biea+bi=eaebi=ea(cosb+isinb)complex e-function is periodical in 2π

Re(ez)=eacosb;Im(ez)=easinb;b=0ez=eao.k.;a=0ez=eib=cosb+isinbo.k.;

Example:
Has the equation ez=1 any solution? (see also 3.5)

z-realnoz-complexeacosb=1easinb=0b=nπeaa=0cosnπn=2k+1=1z=(2k+1)πieπi+1=0beautiful expression


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Similar to the definition of the cos and sin function we have

Definition 7 hyperbolic functions

coshx=12(ex+ex)Rsinhx=12(exex)tanhx=sinhxcoshx=exexex+ex


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like tan x Definitions also valid for complex arguments sinhz=12(ezez),coshz=12(ez+ez),sinh(ix)=12(eixeix)=isinx,cosh(ix)=12(eix+eix)=cosx

Theorem for cosh and sinh:

cosh2xsinh2x=(coshx+sinhx)(coshxsinhx)=exex=1

sinz,cosz for complex arguments are also defined in a logical way:

Definition 8

sinz=12i(eizeiz)cosz=12(eiz+eiz)zC

e.g.: sinz=2=12i(eizeiz)4i=eizweizw24iw1=0w1/2=2i±5iw=eiz=(2±5)iRe(z)=0cosb=0b=(n+12)πsinb=±1±ea=(2±5)a=ln|2±5|z=ln|2±5|+i(n+12)π

The above relation between sin, cos, sinh, and cosh allow e.g. to easily apply the addition theorems to calculate cosh(a+ib)=cosh(a)cosh(ib)+sinh(a)sinh(ib)=cosh(a)cos(b)+isinh(a)sin(b)


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© J. Carstensen (Math for MS)