Complex
Analysis. FreeTutorial
The real sine and cosine
functions are bounded:
Sin(x)1,
Cos(x)1
The
complex sine and cosine functions are
not bounded if they are
defined over the set of all complex numbers.
Complex sine (cosine) functions could
be equal to 2 or 5.
Let us find
such complex
z that
Sin(z)
= a
where a
is real and
a >1.
For example
Sin(z)
= 2
or
Sin(z)
= 5.
Sin(z)
= =
a
(1)
2ia
(2)
Multiply (2)
Let ,
(3)
then
We consider a
is real and
a >1.
It means that
and
iz
= Ln
z
=
=
=
=
=
Ln(i)=
i(
1+2k),
k
= 0, ±1, ±2, ...
z
=
So, when
a = 2
z
=
=
1.7320;
Ln(3.7320) = 1.3169;
Ln(0.2680) = 1.3169
Thus the equation Sin(z)
= 2
has infinitely many solutions
z
= ±
1.3169i,
where k
= 0, ±1, ±2, ...
None of these solutions is a real number.
by
Tetyana Butler
