values of complex cosine
the equation may be written as
This is a quadratic equation in , whence we obtain the two roots (http://planetmath.org/Equation)
The product of the roots is 1, and therefore the roots are distinct from zero for all values of . If we set
the other root is the inverse number
The solution of the equation
is then obtained by taking the complex logarithm
and the other solution of (2) is
In the period strip (1), we have one solution and one solution , both obtained with the value (except in the case with ). In (1), the points and are situated symmetrically with respect the origin. In the cases and , the equation (2) has double roots and , respectively; then we may say that and coincide. Anyhow, we have the
Theorem. In every period strip, cosine attains any complex value at two points.
Example. The solution of the equation is obtained from . In the period strip (1) we get
Since are inverse numbers of each other, we have as result the purely imaginary numbers
From trigonometry, we know that the real zeros of cosine are the odd multiples of ; from these points, belong to the period strip (1). Thus are the only points of (1) where the cosine vanishes. Therefore, according to the preceding theorem, the well-known points
are the only zeros of the cosine function on the whole complex plane.
The values of complex cosine function may be transferred to the complex sine function by means of the complement formula
One can think all points of the -plane to bear the corresponding value of cosine, and then one can translate the plane in the direction of the real axis the distance ; then the values of the sine have been placed to their correct . So one has transferred also the above properties of cosine to sine.
- 1 Ernst Lindelöf: Johdatus funktioteoriaan. Second edition. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1936).
- 2 R. Nevanlinna & V. Paatero: Funktioteoria. Kustannusosakeyhtiö Otava, Helsinki (1963).
|Title||values of complex cosine|
|Date of creation||2013-03-22 17:36:29|
|Last modified on||2013-03-22 17:36:29|
|Last modified by||pahio (2872)|