Lecture 22 Complex Analysis

Hyperbolic function:-
The hyperbolic sine and cosine of complex univariate as real variables. That is
sinh?Z=(e^Z-e^(-Z))/2 , cosh?Z=(e^Z+e^(-Z))/2
tanh?Z=sinh?Z/cosh?Z =(e^Z-e^(-Z))/(e^Z+e^(-Z) )

coth?Z=cosh?Z/sinh?Z =(e^Z+e^(-Z))/(e^Z-e^(-Z) )
? sec?Z??==1/cosh?Z , csc?Z=1/sinh?Z

Characteristics:-
Differentiation:-
d/dZ (sinh?Z )=cosh?Z d/dZ (cosh?Z )=sinh?Z
d/dZ (tanh?Z )=sech^2?Z d/dZ (coth?Z )=-csch^2?Z
d/dZ (sech?Z )=-sech?Z tan?Z
d/dZ (csch?Z )=-csch?Z coth?Z
Relations:-
cosh^2?Z-sinh^2?Z=1
sinh?(Z_1±Z_2 )=sinh??Z_1 ? cos??Z_2 ?±cos??Z_1 ? sinh??Z_2 ?
cosh?(Z_1±Z_2 )=cosh??Z_1 ? cos??Z_2 ?±sinh??Z_1 ? sinh??Z_2 ?
sinh?(-Z)=-sinh?Z
cosh?(-Z)=cosh?Z
sinh?(iZ)=i sin?Z
cosh?(iZ)=cos?Z
sin?(iZ)=i sinh?Z
cos?(iZ)= cosh?(Z)
sinh?Z=sinh?x cos?y+i cosh?x sin?y
SOL:-
sinh?Z=sinh?(x+iy)=sinh?x cosh?iy+cosh?x sinh?iy
=sinh?x cos?y+cosh?x (i sin?y )
?(sinh?Z=sinh?x cos?y+i cosh?x sin?y )
cosh?Z=cosh?x cos?y+i sinh?x sin?y H.W
|sinh?Z |^2=sinh^2?x+sin^2?y
SOL:-
|sinh?Z |^2=|sinh?x cos?y+i cosh?x sin?y |^2
=sinh^2?x cos^2?y+cosh^2?x sin^2?y
=sinh^2?x (1-sin^2?y ) +(1+sinh^2?x ) sin^2?y
=sinh^2?x-sinh^2?x sin^2?y+sin^2?y+sinh^2?x sin^2?y
=sinh^2?x sin^2?y
|cosh?Z |^2=sinh^2?x+cos^2?y H.W

sinh?(Z+2?i)=sinh?Z
SOL:-
Sinh?(Z+2?i)=1/2 [e^(Z+2?i)-e^(-Z-2?i) ]
=1/2 [e^Z?e^2?i-e^(-Z)?e^(-2?i) ]
=1/2 [e^Z (cos?2?+i sin?2? ) e^(-Z) (cos??-2??+i sin??-2?? ) ]
=1/2 [e^Z (1+i0)-e^(-Z) (1+i0) ]=1/2 [e^Z-e^(-Z) ]=sinh?Z
tanh?(Z+?i)=tanh?Z H.W
EX(1):- Prove that |tanh??(1+i)?/4? |=1
Sol:-
|tanh??(1+i)?/4? |=|(e^((1+i)?/4) - e^(- (1+i)?/4))/(e^((1+i)?/4)+? e?^(- (1+i)?/4) )|=|(e^(?/4) ? e?^((-i?)/4) - e^(- ?/4) e^((-i?)/4))/(e^(?/4) ? e?^(i ?/4) + e^(- ?/4) ? e?^(-i?/4) )|
=|(e^(?/4) (cos???/4+i sin???/4? ? )-e^(- ?/4) (cos???/4+i sin???/4? ? ))/(e^(?/4) (cos???/4+i sin???/4? ? )-e^(- ?/4) (cos???/4+i sin???/4? ? ) )|
=| (1/?2 ? e?^(?/4) +i/?2 ? e?^(?/4) +1/?2 ? e?^(- ?/4) +i/?2 ? e?^(- ?/4))/(1/?2 ? e?^(?/4) +i/?2 ? e?^(?/4) +1/?2 ? e?^(- ?/4) +i/?2 ? e?^(- ?/4) ) | =|(1/?2 (? e?^(?/4)-? e?^(- ?/4) )+i/?2 (? e?^(?/4)-? e?^(- ?/4) ))/(1/?2 (? e?^(?/4)-? e?^(- ?/4) )+i/?2 (? e?^(?/4)-? e?^(- ?/4) ) )|=1