8.3 Functions of a Complex Variable
8.3.6 Trigonometric Functions
Having now studied the exponential function we will quickly take a look at thetrigono- metric functions, cosine and sine, which we can easily derive with help of the Euler formula from the exponential function, or define through their power series:
Figure 8.12: Illustration of the horizontal fundamental stripe in the z-plane and the cut w-plane for the exponential function
cosine: w= cosz = eiz+e−iz
2 =
∞
P
n=0
(−1)n z2n
(2n)! and sine: w= sinz = i(e−iz−eiz)
2 =
∞
P
n=0
(−1)n z2n+1 (2n+ 1)!.
Both series converge in the entire complex plane. As we know, cosine and sine are 2π -periodic: cos(z+ 2π) = cosz and sin(z+ 2π) = sinz.
Like our old trigonometric addition theorems from Section 4.2.2:
cos(z±w) = coszcosw∓sinzsinw sin(z±w) = sinzcosw±coszsinw and moreover
cos2z+ sin2z = 1, cos2z−sin2z = cos 2z and 2 coszsinz = sin 2z
hold true for general complex variablesz, w ∈C.
Especially for z+w=x+ iy we obtain from these equations
cos iz = coshz resp. sin iz = i sinhz :
cos(x+ iy) = cosxcos iy−sinxsin iy= cosxcoshy−i sinxsinhy sin(x+ iy) = sinxcos iy+ cosxsin iy= sinxcoshy+icosxsinhy.
The definition of the hyperbolic functions is just as before
hyperbolic cosine: w= coshz = ez+e−z
2 =
∞
P
n=0
z2n (2n)! and hyperbolic sine: w= sinhz = ez−e−z
2 =
∞
P
n=0
z2n+1 (2n+ 1)!.
From this relations we see that, for complex arguments, cosz and sinz are by no means bounded as they are for the real numbers, but rather increase for growing imaginary parts like the hyperbolic functions. Differently from the exponential function, vertical stripes of the z-plane with width 2π, e.g. the fundamental area with −π < Rez ≤ π, is here mapped on the two-sheeted w-plane cut between −1 and +1.
Exercise 8.9 Addition theorems:
Prove one of the addition theorems, e.g. cos(z−w) = coszcosw+ sinzsinw with help of the exponential functions and then show that cos2z+ sin2z = 1.
Exercise 8.10 Connection with the hyperbolic functions
Show that: a) cos iz= coshz, b) sin iz = i sinhz and c) 4 sin3α = 3 sinα−sin 3α.
Exercise 8.11 Functional values of the cosine
Calculate the following functional values of the cosine function: cos±π2, cos±π, cos±iπ2, cos±iπ, cos(π2 ±iπ2), cos(π2 ±iπ), and cos(π±iπ).
In the case of the complex sine function we will demonstrate the wide variety of repre- sentation possibilities which are at our disposal. Because of the symmetry properties it is sufficient to visualize sinz over the square0< x < π and 0< y < π:
The following figures show thelevel curvesfor the real part Re sinz,the imaginary part Im sinz (dashed), the absolute value |sinz| and the argument arg sinz (also dashed) of the mapped functionw= sinz over the square.
Figure 8.13 a + b: Level curve representation for Re sinz and Im sinz over the selected square 0<Rez < π and 0<Imz < π .
Figure 8.13 c + d: Level curve representation for |sinz| and arg sinz over the selected square 0<Rez < π and 0<Imz < π .
Usually these representations are put together in pairs into one single diagram to form a level netas we have done in the next two figures:
Figure 8.13 e + f: Level net for Re sinz and Im sinz and resp. |sinz| and arg sinz over the square
It requires some effort to get an impression of the represented function from the level
curves of the image points. We succeed a little better if correlated with the mean value of the function in that region, the areas between the curves are tinted in grey shades over a scale which extends from deep black for small values to white for large values.
This kind of representation is demonstrated by the Figures g) till j). Now, we can more easily imagine how the values of the imaginary part Im sinz with growing Imz one the one hand increase near Rez = 0 and on the other hand decrease near Rez =π. Also the tremendous increase of Re sinz and |sinz|with growing distance from the real axis shows up clearly.
Figure 8.13 g + h: Grey tinted level curve representation for Re sinz and Im sinz over the square.
Figure 8.13 i + j:Grey tinted level curve representation for |sinz| and arg sinz over the square.
We achieve even more impressive pictures, if we use a colour scale to characterize the
relative heights, for instance as done in geographical maps with their deep blue of the deep oceans, different shades of green representing low lying land, the beige, and finally the darker and darker browns of the mountains. In the computer program MATHEMATICA used here in the Figures k) till n) the colours of the rainbow are used to represent function height according to the frequency of the light. (Magma-)Red colours represent smaller values and (sky-)blue colours stand for higher functional values. Through these figures we get a significant impression of the structure of the “function value mountains”.
Figure 8.13 k + l: Rain-bow-like coloured level curve representation for Re sinz and Im sinz over the square.
Figure 8.13 m + n: Rain-bow-like coloured level curve representation for |sinz| and arg sinz over the square.
In figure 8.13 n we see particularly well the linear increase of the phase from −90◦ at Rez =π to +90◦ at Rez = 0.
Also with this kind of representation, the coloured visualized level curves of a variable can be upgraded to anetthrough the marking in of the (dashed) level curves of a second variable which however cannot yet be represented by colours. This fact is illustrated in the next two figures:
Figure 8.13 o + p: Rain-bow-like coloured level net representation for Re sinz and Im sinz and resp. |sinz| and arg sinz over the square.
We get, however, a more vivid impression than these two-dimensional projections are able to provide, if we look at the pictures of the function values in perspective which are offered by the drawing programs of the modern computers as shown in the next figures:
Figure 8.14 a + b: Relief picture in perspective of the function values of Resinz and Im sinz with an x-y net over the selected square 0<Rez < π and 0<Imz < π .
Figure 8.14 c + d: Relief picture in perspective of the function values of |sinz| and arg sinz with an x-y net over the selected square 0<Rez < π and 0<Imz < π.
To demonstrate the influence of the changes in sign, we have displayed for you finally