Complex Trigonometric Functions: Sine, Cosine, Tangent

7.4 Complex Trigonometric Functions: Sine, Cosine, Tangent

From the complex exponential function it is only on step to the complex trigonometric function. In addition to the Euler formula $e^{i z} = cos z + i sin z$ we require the definitions of the hyperbolic functions $sinh$ and $cosh$ :

\begin{matrix} e^{i z} = cos z + i sin z; e^{- i z} = cos (- z) + i sin (- z) = cos z - i sin z; \to \\ sin z = \frac{e^{i z} - e^{- i z}}{2 i}; cos z = \frac{e^{i z} + e^{- i z}}{2}; \\ sinh z = \frac{e^{z} - e^{- z}}{2}; cosh z = \frac{e^{z} + e^{- z}}{2}; \to {cosh}^{2} z - {sinh}^{2} z = 1; \\ auxiliary results : \\ cos z = cosh (i z); sin z = (1 ∕ i) sinh (i z); cos (i z) = cosh (z); sin (i z) = i sinh z \end{matrix}

\begin{matrix} With e^{i z} = e^{i x - y} = e^{- y} e^{i x} = e^{- y} (cos x + i sin x) \\ e^{- i z} = e^{- i x + y} = e^{y} e^{- i x} = e^{y} (cos x - i sin x) \\ it follows that sin z = sin x \frac{(e^{y} + e^{- y})}{2} + i cos x \frac{(e^{y} - e^{- y})}{2} = sin x cosh y + i s i cos x sinh y \\ cos z = cos x \frac{(e^{y} + e^{- y})}{2} - i sin x \frac{(e^{y} - e^{- y})}{2} = cos x cosh y - i sin x sinh y \\ tan z = \frac{sin z}{cos x} = \frac{sin x cosh y + i s i cos x sinh y}{cos x cosh y - i sin x sinh y} = \frac{(sin x cosh y + i s i cos x sinh y) (cos x cosh y + i sin x sinh y)}{(cos x cosh y - i sin x sinh y) (cos x cosh y + i sin x sinh y)} \\ tan z = \frac{sin x cos x + i sinh y cosh y}{{cos}^{2} x + {sinh}^{2} y} \end{matrix}

7.4.1 Complex Sine

When shifting the point arrays parallel to the real axis one observes their periodic mapping. The square array is then mapped into a region that is bounded by orthogonal ellipses and hyperbolas. Further details and hints for experiments are given in the description pages of the simulation.

Figure 7.7: Conformal of a point grid and a circular array around the origin with radius

π ∕ 2

from the

z

-plane to the

u

-plane. The circle with radius

π ∕ 2

in the

z

-plane and the unit circle in the

u

-plane are drawn in black. In the

z

-plane the boundaries of a period are drawn in red. The Play button shifts the square array along the real axis.

As is to be expected the mapping via the cosine for a phase shift by $π ∕ 2$ on the real axis leads to the same result as the mapping for the sine. Fig.7.8 shows this for the same configuration of the $u$ -plane as in Fig.7.7 with $u = sin z$ . Further details and hints for experiments are given in the description pages of the simulation.

Figure 7.8: Conformal mapping of a point grid that has been shifted by

π ∕ 2

relative to the origin to the

u

-plane. The circle with radius

π ∕ 2

in the

z

-plane and the unit circle in the

u

-plane are drawn in black. In the

z

-plane the boundaries of a period are drawn in red.

7.4.2 Complex Tangent

In addition to the expected periodicity under shifts parallel to the real axis, the complex tangent $u = tan z$ shown in Fig.7.8 shows a wealth of interesting phenomena because of its divergence with sign change at odd multiples of $π ∕ 2$ . Because of the large sensitivity close to the divergences you should use the two number fields to enter exact values for $x$ and $y$ . They can be chosen outside of the intervals covered by the sliders.

Figure 7.9: Conformal mapping of square point grid and a circular array around the origin of the

z

-plane to the

u

-plane. A circle with radius

π ∕ 2

and in the

z

-plane and the unit circle are drawn in black. Parallel to the imaginary axis the boundaries of a period are drawn in red. Play shifts the square array parallel to the real axis.

Straight lines parallel to the real land imaginary axis here are mapped into closed curves around and through the points $+ i$ and $- i$ . The region with imaginary value large $π$ is mapped to the point $i$ , the region with imaginary values smaller $π$ is mapped to the point $- i$ . Further details and hints for experiments are given in the description pages of the simulation.