Complex trigonometric functions
Euler´s Formula: e iz = cosz + i sinz; e -iz =
cosz - i sinz
Definition: cosh z = (ez + e-z)/2; sinh z
= (ez + e-z)/2 ➙ cosh2z -
sinh2z = 1
Complex trigonometric function
e z = e x + iy = ex (cos y + i sin
y); e -z = e x - iy = ex (cos y - i sin
y);
➙ sin z = 1/2i (e iz - e -iz) = sin x
cosh y + i cos x sinh y
cos z = 1/2 (e iz +e -iz) = cos x cosh y - i
sin x sinh y
= sin (x + π/2) cosh y + i cos (x + π/2) sinh y
tan z = sin z / cos z = (sin x cosh y + i cos x sinh y) / (cos
x cosh y - i sin x sinh y)
Multiplying nominator and denominator with conjugate complex,
and cosh2z - sinh2 z = 1
➙ tan z = (sin x cos x + i sinh y cosh y) / (cos2 x
+ sinh2 y)