The complex sine function is, as in the real case, defined as the solution of the
differential equation (ODE) sin''(z) = -sin(z)
to the initial conditions sin(0) = 0, sin'(0) = 1
. The
real and the complex sine function therefore agree for real arguments x. The function g(z) = sin(i*z)
satisfies the differential equation g''(z) = +g(z) with g(0) = 1, g'(0) = i
. By definition of the
hyperbolic sine as solution of this ODE we conclude sin(i*z) = i*sinh(z)
.
z → f(z) = sin(c*z)/c
-1.57 ≦ Re(z) ≦ 1.57; -2 ≦ Im(z) ≦ 2
1 ≥ c ≥ 0
: as c approaches 0 the function converges to the linear function z → z.exp(z) = cosh(z) + i*sinh(z) = cos(i*z) + sin(i*z)
, and also cos(z) = sin(z + pi/2)
.
Therefore one can read the values of sin, cos, sinh, cosh from this one picture - in the same way
as the graph of the real sine function also gives the values of the real cosine function.f(z)
where f'(z) = 0
. Of course sin'(pi/2) = 0 = sin'(-pi/2)
as in the real case.
z → f(z) = sin(c*z)/c
-pi/2 + b ≦ Re(z) ≦ pi/2 + b ; 0 ≦ Im(z) ≦ 2.5
sin(z) = sin(z+2pi)
.sin(z)
grows fast for z = i*y
on the imaginary axis (recall sin(i*y) = i*sinh(y)
) we can draw
only a small portion of the fundamental domain {z ; -pi ≦ Re(z) ≦ pi} for the period translations.
z → f(z) = sin(c*z)/c
0.5 ≦ |z| ≦ pi/2, 0 ≦ angle ≦ 2*pi
1 >= c >= 0
.