lograthmic

hyperbolic function:
the hyperbolic sine ,cosine , tan and cot for complex number:
sinhz=(ez-e-z)/2
coshz=(ez+e-z)/2
tanhz=( ez-e-z)/( ez+e-z)
cotchz=( ez+e-z)/( ez-e-z)
1-cosh(iz)=cosz
2-sinh(-z)=-sinhz
3-cosh(-z)=coshz
4-Sinh(z1+z2)=sinhz1coshz2+coshz1sinhz2
Proof:
R.H.S=((( ez1-e-z1)/2).( ez2+e-z2)/2)+((( ez1+e-z1)/2).( ez2-e-z2)/2)
= (ez1+z2 +ez1-z2-e-z1+z2-e-(z1+z2)+ez1+z2-ez1-z2+e-z1+z2-e-(z1+z2)) /4
= (2 e(z1+z2)-2 e-(z1+z2))/4
=( ez1+z2- e-(z1+z2))/2=sinh(z1+z2)
5-tan(iz)=itanhz
6-d/dz tanhz=sech2z
Proof:
d/dz tanhz=d/dz ( ez-e-z)/( ez+e-z)= d/dz(e2z-1)/(e2z+1) (multiply by ez/ez )
=((e2z+1)(2e2z)-(e2z-1).2.e2z)/(e2z+1)2
=4e2z/(e2z+1)2
=(2ez/e2z+1)2
=(2/(ez+e-z)2
=(1/coshz)2
=sech2z
7-d/dz cothz=-cosech2z
8-cot(iz)=-icothz
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?0)? ?