logarithmic function
logarithimic function is defined at non-zero complex point
z=r.exp??(i?)=re^i? ?
log??(z)=log??(r.e^i? )=log??r+i? ( as log??e^i?=i? log??(e)=i?? ? )? ? ?
or ? log???z=lnr+i? ,r>0 -?z=re^i?=r(cos?+isin ?)
=r(cos??(?±2k?)+isin(?±2k?))?=re^(i(?±2k?))=r exp??(i(?±2k?))?
log??z=log??r+i (?±2k?)? ,k=0,1,2,3,…?
or log??z=ln??r+i(?±2k?)……(1) k=0,1,2,3,…? ?
principls value of log??z ?
put k=0 ,in equation(1)
log??z=ln??r+i?? ?
log??z=log??|z|+i Arg??(z)? ? ?
now that log??z=log??r ±2k?i , k=0,1,2,…? ?
z=r.e^i?,? has any one of the value ?=?+2k?
log??z=ln??r+i?…(2)? ?
that is log??z=ln??|z|+i arg?z (z?0)? ?
properties logarithmic?? of complex number?-?
1-exp??(log?z )=z?
2-log??(e^z )=z+2k?i ,k=0,±1,±2,…?
3-log??(1)=2k?i k=0,±1,±2,…?
4-log??(i)=(2k+1/2)?i k=0,±1,… ?
5-log??e=1+2k?i k=0,±1,…?
6-log??(-1)=(2k+1)?i k=0,±1,…?
?7-log???(z_1/z_2 )=log??z_1-log??z_2 ? ? ?
8-log??(z_1 z_2 )=log??z_1+log??z_2 ? ? ?