2.3 Complex e-function

Function: f(x) =ex ,x , 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 +x n ) n=n=0 1n !xn Euler’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 e xey =n =0x nn! m=0 ymm != n=0 k=0n xkyn-kk !(n -k)! =n=0 1n!k=0n nkxk yn-k = n=0 (x+ y)nn ! =ex +y

Euler’s formula:
taking into account i4k =1,i 4k+1 =i,i 4k+2 =-1 , i4 k+ 3=-i , and using the definitions by the Taylor series we find eiφ =n=0 (iφ)n n! =1+iφ +(iφ)22!+ (iφ ) 33 ! + (i φ)4 4! + (iφ) 55! + =1- φ22 !+φ 44! ++i ( φ-φ 33!+φ5 5!- ) :=cosφ +i sinφ

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.

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 e ix =cosx+i sinx x e -ix =cos x-i sinx ⇒  cosx =1 2 ( eix+ e-ix ) sinx =12i ( eix- e-ix)

Additionally we directly get

cos2x+ sin2x = (cosx+i sinx ) ( cosx-i sinx )=e i xe- ix=1

Addition theorems for sin and cos functions:
Combining the exp-addition formula with Euler’s formula we find (cosy cos z- siny sinz )+i (cosy sinz+ siny cosz ) = (cosy +i siny) (cosz +i sinz) =eiy eiz =ei (y+z) =cos (y+z )+ i sin( y+z )

From real and imaginary part we finally get (representing the even and odd part of the complex exponential function) cosy cosz - siny sinz =cos (y+z ) cos y sinz + siny cosz=sin( y+ z)

Combining both equations we easily get

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

Back to complex numbers:


→ In general:
z=r (cosφ +i sinφ ) Re {z}=r cosφ Im {z} =r sinφr =|z |

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Multiplication of complex numbers: z 1z 2 =r 1eiφ 1r 2eiφ 2 =(r 1r2 )ei ( φ1 +φ 2 )= (r1r 2 )(cos(φ 1+φ 2)+i sin( φ1+φ 2) ) zz =rei φre -iφ=r 2

Definition 6

f(z) =ezz is the complex e -function with
z =a+bi⇒ ea +bi=e aebi=ea(cosb+i sinb )⇒ complex  e-function is periodical in  2π

Re(e z) =e a cosb ; Im (e z ) =e a sinb ; b =0 → ez =ea o.k.;a =0→ ez =eib = cosb +i sinbo.k. ;

Example:
Has the equation ez =-1 any solution?

z-real → no z-complex⇒  ea cosb= -1  (see 3.5) ea sinb=0⇒ b =nπ ⇒  ea a=0 cosn πn =2k+1 =-1 ⇒ z=(2k+1)πi→  e πi+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 (e x +e-x) sinhx=1 2 ( ex - e-x) tanhx = sinhx coshx =e x -e -x ex+ e -x


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like tan x →  Definitions also valid for complex arguments sinhz=12 (ez -e- z) , coshz=12 (e z+e-z), sinh (ix)=12 (e ix -e -ix) =i sinx , cosh ( ix ) = 12 (eix+e-i x )= cosx

Theorem for cosh and sinh:

cosh2x - sinh2x = (coshx+ sinhx) (coshx- sinhx) =exe-x=1

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

Definition 8

sinz =1 2i ( eiz -e-iz ) cosz=1 2 ( eiz + e-iz ) z

e.g.: sinz=2=1 2i ( eiz -e-iz ) ⇒ 4i = eiz w-e-iz w2-4 iw -1=0⇒  w12 = 2i 5i w=eiz =(25 )i → Re(z )= 0→  cosb=0→  b= ( n+12 )π →  sinb=1 ea = (2 5) a =ln |25 | ⇒ 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)+i sinh (a ) sin (b)


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© J. Carstensen (Math for MS)