2.4.1 Conformal Mapping

Analytic functions may be defined in several equivalent ways. From the viewpoint of geometry, the defining characteristic of analytic functions is their angle-preserving property. This property will now be examined in some detail.

Let \(\zeta=f(z)\) be an analytic function and suppose that \(f^\prime(z_{0}) \neq 0\) for some point \(z_{0}\) of its domain of definition. Then, according to third theorem of the previous the function \(f (z)\) provides a continuous one-to-one mapping of the neighborhood of \(z_{0}\) onto the neighborhood of its image \(\zeta_{0}=f(z_{0})\) . Now consider any smooth curve \(C\) passing through \(z_{0}\) and let \(C\) be described in parametric form by differentiable functions \[x=x(t); \quad y=y(t)\] or equivalently by \[z(t)=x(t)+i y(t)\] with \(z_{0}=z(t_{0})\) . The direction angle of the curve at the point \(z_{0}\) is the angle \(\theta\) made by the tangent to the curve at \(z_{0}\) with the real axis. Thus \[\tag{4.10} \tan \theta=\frac{{dy}}{{dx}}=\frac{\dot{{y}}({t}_{0})}{\dot{{x}}({t}_{0})}\] and if we write \[\dot{z}(t_{0})=\dot{x}(t_{0})+i \dot{y}(t_{0})\] we have simply \[\tag{4.11} \theta= \operatorname{am}\big(\dot{z}(t_{0})\big).\] 

In the neighborhood of \(\zeta_{0}\) the image of \(C\) is a curve \begin{align} C': \quad \zeta(t)&=f[z(t)]\\ &=u[x(t), y(t)]+i v[x(t), y(t)]. \end{align} We conclude that \begin{align} \dot{\zeta}(t) & =u_{x} \dot{x}+u_{y} \dot{y}+i\left(v_{x} \dot{x}+v_{y} \dot{y}\right)\\ & =\left(u_{x}+i v_{x}\right)(\dot{x}+i \dot{y})\\ &=f'(\dot{z}) \dot{z}(t) \end{align} so that the direction angle of the curve \(C'\) at \(\zeta_{0}=f(z_{0})\) is \[\tag{4.12} \theta' = \operatorname{am}\big(f'(z_{0}) \dot{z}(t_{0})\big).\] Now let us consider two curves \(z_{1}(t)\) and \(z_{2}(t)\) meeting at an angle \(\alpha\) at \(z_{0}\) and let us see what can be said for the angle \(\beta\) of intersection of their respective images \(\zeta_{1}(t)\) and \(\zeta_{2}(t)\) at \(\zeta_{0}\) . From (4.11) we have \begin{align} \alpha=\theta_{2}-\theta_{1}&=\operatorname{am}\big(\dot{z}_{2}(t_0)\big)-\operatorname{am}\big(\dot{z}_{1}(t_0)\big)\\ &=\operatorname{am}\bigg(\frac{\dot{z}_{2}(t_{0})}{\dot{z}_{1}(t_{0})}\bigg) \end{align} and from (4.12) \begin{align} \beta=\theta_{2}'-\theta_{1}'&=\operatorname{am}\bigg(\frac{f'(z_{0}) \dot{z}_{2}(t_{0})}{f'(z_{0}) \dot{z}_{1}(t_{0})}\bigg)\\ &=\operatorname{am}\bigg(\frac{\dot{z}_{2}(t_{0})}{\dot{z}_{1}(t_{0})}\bigg)\\ &=\alpha. \end{align} We have proved for any point where \(f'(z) \neq 0\) that the angle between two curves and also its sense remains unchanged. Such a map is said to be conformal . What occurs at points where the derivative vanishes will be discussed in a later section.

This mapping property is sufficient to define analytic functions. More precisely, if a function \(\boldsymbol{\zeta=f(z)}\) defines a one-to-one conformal map in some domain \(\boldsymbol{D}\) where its real and imaginary parts are differentiable functions of \(\boldsymbol{x}\) and \(\boldsymbol{y}\) then \(\boldsymbol{f(z)}\) is analytic in \(\boldsymbol{D}\) . For proof consider any two curves \(y=y_{1}(x)\) and \(y=y_{2}(x)\) through a given point \(z\) of \(D\) . These meet at an angle \(\alpha\) such that \[\tan \alpha=\frac{y_{2}'(x)-y_{1}'(x)}{1+y_{2}'(x) y_{1}'(x)}.\] These will be mapped by \(f(z)=u(x, y)+i v(x, y)\) onto the curves \[\left\{\begin{array} { l } { u _ { 1 } = u ( x , y _ { 1 } ( x ) ) ; } \\ { v _ { 1 } = v ( x , y _ { 1 } ( x ) ) ; } \end{array} \quad \left\{\begin{array}{l} u_{2}=u\left(x, y_{2}(x)\right) \\ v_{2}=v\left(x, y_{2}(x)\right). \end{array}\right.\right.\] By our assumption \[\tan \alpha=\frac{\frac{d v_{2}}{d u_{2}}-\frac{d v_{1}}{d u_{1}}}{1+\frac{d v_{2}}{d u_{2}} \frac{d v_{1}}{d u_{1}}}.\] Thus, using \(\frac{d v_{1}}{d u_{1}}=\frac{v_{x}+v_{y} y_{1}'}{u_{x}+u_{y} y_{1}'}\) and a similar expression for \(\frac{d v_{2}}{d u_{2}}\) we obtain \[\frac{y_{2}'-y_{1}'}{1+y_{2}' y_{1}'}=\frac{\left(u_{x} v_{y}-u_{y} v_{x}\right)\left(y_{2}'-y_{1}'\right)}{\left(u_{x}^{2}+u_{y}^{2}\right)+\left[u_{x} u_{y}+v_{x} v_{y}\right]\left(y_{1}'+y_{2}'\right)+\left[v_{x}^{2}+v_{y}^{2}\right] y_{1}' y_{2}'}\] an equation which must hold for all values of \(y_{1}'\) and \(y_{2}'\) . Therefore we may multiply to cancel the denominators and then equate coefficients of corresponding terms. We then obtain the relations \[\tag{4.13} u_{x} u_{y}+v_{x} v_{y}=0\] and \[\tag{4.14} u_{x}^{2}+u_{y}^{2}=u_{x} v_{y}-u_{y} v_{x}=v_{x}^{2}+v_{y}^{2}\] where, by our assumption \(u_{x} v_{y}-u_{y} v_{x} \neq 0\) . Equation (4.13) may be rewritten \[\tag{4.15} \frac{u_{x}}{v_{y}}=-\frac{v_{x}}{u_{y}}=\lambda.\] The equation (4.14) then takes the form \[v_{y}^{2} \lambda^{2}+u_{y}^{2}=\lambda\left(v_{y}^{2}+u_{y}^{2}\right)=u_{y}^{2} \lambda^{2}+v_{y}^{2}\] whence \begin{align} \left(\lambda^{2}-\lambda\right) u_{y}^{2}+(1-\lambda) v_{y}^{2}&=0 \\ (1-\lambda) u_{y}^{2}+\left(\lambda^{2}-\lambda\right) v_{y}^{2}&=0 \end{align} or \[\left(\lambda^{2}-2 \lambda+1\right)\left(u_{y}^{2}+v_{y}^{2}\right)=0.\] Since the second factor cannot be zero without making the Jacobian vanish we can only have \(\lambda=1\) . From (4.15) we conclude that \begin{align} u_{x}&=v_{y} \\ u_{y}&=-v_{x}. \end{align} But these are the familiar Cauchy-Riemann equations so that \(f(z)\) is, in fact, an analytic function.

Since angles are preserved by the mapping we see that an analytic function maps sufficiently small triangles into nearly similar figures. In other words, the transformation of the neighborhood of a point \(z_{0}\) is just an ordinary magnification by a constant factor which depends only on \(z_{0}\) . To verify this consider the curve \(C\) through \(z_{0}\) and its image \(C'\) and let \(s\) and \(\sigma\) denote their respective arclengths. Then \begin{align} \frac{{d} \sigma}{{ds}}=\frac{{d} \sigma}{{d} t} \Big/ \frac{{ds}}{{d} t}&=\frac{\sqrt{\dot{{u}}^{2}(t_{0})+\dot{{v}}^{2}(t_{0})}}{\sqrt{\dot{x}^{2}(t_{0})+\dot{{y}}^{2}(t_{0})}}\\ &=\frac{\left|\frac{{dw}}{{dt}}\right|}{\left|\frac{{dz}}{{dt}}\right|}\\ &=\left|{f}'({z}_{0})\right|. \end{align} So the length of any linear element passing through \(z_{0}\) is simply magnified by the factor \(\left|f'(z_{0})\right|\) . The significance of the requirement that the derivative of an analytic function be the same for any direction of approach should now be quite clear.

2.4.2 The Mapping \(\boldsymbol{w=z^{n}}\) 

We have already considered the mapping properties of the general linear function. After that the simplest example of an analytic function is \(w=z^{n}\) , where \(n\) is a positive integer. The function is differentiable in the entire \(z\) -plane with the derivative \(w'(z)=n z^{n-1}\) . Since \(w'(0) \neq 0\) anywhere but the origin we conclude that the mapping is conformal everywhere except perhaps for the one point \(z=0\) . Using the polar form \(z=r e^{i \theta}\) and \(w=\rho e^{i \phi}\) (after (3.23) ) we obtain the relations \[\tag{4.21} \rho=r^{n}; \quad \phi=n \theta.\] We conclude that circles about the origin map into circles about the origin and rays emanating from the origin map into rays emanating from the origin. In the relation \(\phi=n \theta\) we observe the special nature of the mapping. The angle between two rays meeting at the origin is not preserved, but multiplied \({n}\) -fold. In other words, the mapping at the origin instead of being conformal has the property of multiplying angles by \(n\) . A point which has this property is called a branch point of order \(\boldsymbol{n - 1}\) .

The circles \(r= \textit{const.}\) go into the circles \(\rho= \textit{const.}\) so that a circular sector with vertex at the origin will be mapped into a circular sector with \({n}\) times the central angle. In particular, the function \(w=z^{n}\) provides us with a conformal map of the interior of a wedge with vertex angle \(\frac{\pi}{n}\) onto the entire upper half-plane.

When we attempt to consider the inverse mapping we encounter a new difficulty. With the exception of the origin a point of the \(w\) -plane is the image of more than one point of the \(z\) -plane. For instance, the mapping of the unit circle in the \(z\) -plane covers the unit circle in the \(w\) -plane \({n}\) times. In general, to every point \(w=\rho e^{i \phi} \neq 0\) of the \(w\) -plane there correspond \(n\) distinct points in the \(z\) -plane, the points with \(r=\rho^{1 / n}\) and \(\theta=\frac{\phi+2 k \pi}{n}\) , \((k = 0, 1, \ldots, n-1)\) . The inverse function \[\tag{4.22} z=\sqrt[n]{w}\] is not uniquely defined. In fact it is possible to assign \(n\) distinct values to \({z}\) for every \(w \neq 0\) . Such a correspondence is called a multivalued function and in this particular instance an \(\boldsymbol{n}\) -valued function . The word function unmodified, unless the context makes the contrary clear, will be used to denote single-valued function.

The values of the function (4.22) are tied together in a very specific way. For example, if we take any neighborhood excluding the origin in the \(w\) -plane and arbitrarily assign any one of the \(n\) possible values to any point of the interior then the assignment of values in the rest of the neighborhood is completely and uniquely fixed by the requirement of continuity alone. By means of such a specific assignment of values we have obtained a one-to-one mapping of part of the \(w\) -plane, although we could have done this in \({n}\) ways. Nonetheless, this treatment suggests that we connect the values of the function by feeling our way, as it were, from neighborhood to neighborhood and tying in those values which keep the function continuous. In this manner we may eventually return to a point with a function value other than the one we began with. We treat the situation as though we were on another "sheet" overlapping the original one. Thus the mapping \({z}=\sqrt[n]{{w}}\) may be considered as one-to-one if we imagine the \(w\) -plane to consist of \(n\) overlapping sheets with a common point at the origin and various interconnections.

This, in an intuitive way, is the general concept of a Riemann surface . It is a surface which allows us a one-to-one representation of a multi-valued function. In this fashion we treat with great simplicity a number of problems which would be difficult to approach without this organizing concept. Let us examine the Riemann surface for a simple function, \(w=\sqrt{z}\) . In the mapping \(w=z^{2}\) any ray, \(\phi=\phi_{0}\) in the \(w\) -plane is the image of two rays of the \(z\) -plane, \(\theta=\frac{\phi_{0}}{2}\) and \(\theta=\frac{\phi_{0} + 2\pi}{2}\) . We start with the positive real axis \(\phi=0\) and assign to it the positive real axis of the \(z\) -plane, \(\theta=0\) . As we let \(\phi\) go from 0 to \(2 \pi\) , \(\theta\) goes from \(0\) to \(\pi\) . In other words as the ray sweeps over the entire \(w\) -plane the corresponding ray in the \(z\) -plane only sweeps over the upper half-plane. The domain consisting of the slit plane \(0<\phi<2 \pi\) is mapped in one-to-one fashion on the half plane \(0<\theta<\pi\) . The mapping of the boundary is not one-to-one, however, since the positive real axis \(\phi=0\) or \(\phi=2 \pi\) corresponds to both rays \(\theta=0\) and \(\theta=\pi\) . Nevertheless, we may distinguish at least formally between the representations \(\phi=0\) and \(\phi=2 \pi\) of the positive real axis and thus obtain a convenient way of distinguishing between the two sheets of the Riemann surface for any point on the real axis. Now, if we continue in conception on a duplicate sheet where \(\phi\) ranges from \(2 \pi\) to \(4 \pi\) we can extend the mapping through the lower \(z\) -plane. To complete the picture of the mapping we now have to connect the two \(w\) -sheets in a manner corresponding to the connection between the lower and upper half \(z\) -planes. That is we now join the lower edge of the slit in the first sheet with the upper edge of the slit in the second sheet for they both correspond to the angle \(\phi=2 \pi\) and are to be identified. Next the edges \(\phi=0\) and \(\phi=4 \pi\) are to be joined to each other for they both correspond to the ray \(\theta=0\) . That this cannot be accomplished mechanically without self-intersecting the surface does not matter here; there is no difficulty in conceiving of such a model abstractly. The resulting surface consisting of two superimposed complex planes winding around the origin is the complete Riemann surface of the function \(z=\sqrt{w}\) . On it \(z\) is defined as a one-valued function of \(w\) , and therefore this surface is mapped by means of \(z=\sqrt{w}\) uniquely on the \(z\) -plane. The two different determinations or "branches" of \(\sqrt{w}\) are represented on the two distinct sheets of the Riemann surface. The points \(w=0\) and \(w=\infty\) which are common to both sheets and at which no differentiation between the two sheets is possible are called branch points of the first order . We might add that the positive real axis does not occupy any preferred position in this discussion. Instead we might have introduced a slit in the \(w\) -plane along any other continuous non-self-intersecting curve joining the two branch points \({w}=0\) , \({w}=\infty\) .

It is quite clear how to proceed in the case \(w=z^{n}\) . There we would obtain a one-to-one mapping of the simple \(z\) -plane onto a Riemann surface of \(n\) sheets winding around the origin with branch points of \((n-1)^{\text{st}}\) order at \(w=0\) and \(w=\infty\) . This is the Riemann surface of \(\sqrt[n]{w}\) .

It is interesting to see what happens to the coordinate lines in the \(z\) -plane in the mapping \(w=z^{2}\) and to the coordinate lines in the \(w\) -plane under the inverse mapping. Setting \(w=u+iv\) and \(z=x+iy\) we have \[w=u+i v=x^{2}-y^{2}+2 i x y\] or \[u=x^{2}-y^{2}, \quad v=2 x y.\] Thus the hyperbolas \(x^{2}-y^{2}=c>0\) are mapped on the lines \(u=c\) , so that the two branches of a hyperbola of this family are mapped on the lines \(u=c\) in each of the two sheets of the Riemann surface. Further, the domain bounded by each of the branches of the hyperbola (shaded in the figure) is mapped on the domain \(u>c\) . The orthogonal family \(2xy=c\) is mapped, as one can verify directly or deduce from the conformality of the mapping, on the lines \({v}={c}\) .

For the image of the coordinate lines \(x=c>0\) we have \(u=c^{2}-y^{2}\) , \(v=2 c y\) . This is a family of confocal parabolas in the \(w\) -plane with focus at \(w=0\) and vertices at \(w=c^{2}\) . Clearly, the exterior of each of these parabolas (i.e., the domain not including the origin) corresponds to the half-planes \(x>c\) . Similarly, one gets for the images of the lines \(y=c\) the orthogonal family of parabolas \[u=x^{2}-c^{2}, \quad v=2 c x.\] Their exteriors are mapped on the half-planes \(y>c\) . Also the images \(x=-c\) and \(y=-c\) are identical with those of \(x=c\) and \({y}={c}\) .

Exercises

2.4.3 The Exponential Function and the Logarithm

For the moment we ignore the definition of Chapter 2, (3.21) and define the exponential function \(e^{z}\) in a way which is perhaps more natural. We set \[\tag{4.31} e^{z}=\sum_{n = 0}^{\infty}\frac{z^{n}}{n !}=1+z+\frac{z^{2}}{2!}+\cdots\] in analogy to the real function \({e}^{{x}}\) . It is easy to verify the fact that this series converges for all finite \(z\) (cf. Chapter 1, (2.22) ). We have proved that a power series is analytic in its circle of convergence and may be differentiated. Hence the series (4.31) defines a function analytic in the entire \(z\) -plane and as in the real case \[\frac{d}{d z}(e^{z})=\sum_{n = 1}^{\infty} \frac{z^{n-1}}{(n-1)!}=e^{z}.\] To demonstrate that the addition formula \[\tag{4.32} e^{z_{1}} e^{z_{2}}=e^{(z_{1}+z_{2})}\] also holds for complex values we multiply the two series term by term (the series are absolutely convergent). We have \begin{align} e^{z_{1}} e^{z_{2}} & =\sum_{n=0}^{\infty} \frac{z_{1}^{n}}{n !} \sum_{n=0}^{\infty} \frac{z_{2}^{n}}{n !} \\ & =\sum_{n=0}^{\infty} \sum_{r=0}^{n} \frac{z_{1}^{r}}{r !} \frac{z_{2}^{n-r}}{(n-r) !} \\ & =\sum_{n=0}^{\infty} \frac{1}{n !} \sum_{r=0}^{n} \frac{n !}{r !(n-r) !} z_{1}^{r} z_{2}^{n-r} \\ & =\sum_{n=0}^{\infty} \frac{1}{n !} \sum_{r=0}^{n}\binom{n}{r} z_{1}^{r} z_{2}^{n-r}. \end{align} 

But, by the binomial theorem this is simply \[\sum_{n=0}^{\infty} \frac{1}{n !}(z_{1}+z_{2})^{n}=e^{(z_{1}+z_{2})}\] which proves our assertion.

Next we prove an important identity of Euler’s which connected the exponential function with the trigonometric functions. We define the functions \(\cos z\) and \(\sin z\) by employing the series for the corresponding real functions with complex arguments. Thus we set \begin{align} \tag{4.33} \sin{z} &= \sum_{n = 0}^\infty \frac{(-1)^n z^{2n+1}}{(2n+1)!} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots\\ \cos{z} &= \sum_{n=0}^{\infty} \frac{(-1)^{n} z^{2 n}}{(2n)!}=1-\frac{z^{2}}{2 !}+\frac{z^{4}}{4 !}-\cdots \end{align} These also converge for all finite values and consequently represent analytic functions in the full plane. It is easy to check that \[\frac{d}{d z} \sin z=\cos z; \quad \frac{d}{d z} \cos z=-\sin z.\] Since the series are absolutely convergent we may write \begin{align} \cos z+i \sin z&=\sum_{n=0}^{\infty}\left(\frac{i^{2 n} z^{2 n}}{(2 n) !}+\frac{i^{2 n+1} z^{2 n+1}}{(2 n+1) !}\right)\\ &=\sum_{n=0}^{\infty} \frac{(i z)^{n}}{n !} \end{align} and thus obtain Euler’s identity \[\tag{4.34a} e^{i z}=\cos z+i \sin z.\] From this and (4.32) we obtain \[\tag{4.34b} e^{z}=e^{x+i y}=e^{x}(\cos y+i \sin y)\] which proves the equivalence of this definition with that of Chapter 2, (3.21) . Furthermore, by virtue of the periodicity of the real trigonometric functions we have \[e^{z+2 \pi i}=e^{z}\] or \(e^{z}\) is periodic with period \(2 \pi i\) .

In order to consider the mapping defined by the exponential function \(w=e^{z}\) , we express \(w\) in polar coordinates \(w=\rho e^{i \phi}\) , \({z}\) in rectangular coordinates, \({z}={x}+ iy\) . We then have \[\rho={e}^{{x}}; \quad \phi={y}.\] From this it follows that the line \({y}={c}\) parallel to the real axis is mapped onto the ray \(\phi=c\) , the ray running from \(0\) to \(\infty\) as \(x\) goes from \(-\infty\) to \(\infty\) . Thus the value of \(e^{z}\) at \(\infty\) is not defined since we have seen it to depend on the direction of approach.

Let us consider what happens to the lines \({x}= \textit{const.} ={c}\) . We may limit ourselves, because of the periodicity of the exponential function, to a segment \(x=c\) and \(0. Clearly this segment will be mapped on the circular arc \(\rho=e^{c}\) , \(0<\phi<\alpha\) . The horizontal strip in the \(z\) -plane bounded by \(y=0\) and \({y}=\alpha<2 \pi\) is mapped on the wedge bounded by the rays \(\phi=0\) and \(\phi=\alpha\) in the \(w\) -plane. Thus a strip of width \(2 \pi\) , i.e., \(0, is mapped on the entire \(w\) -plane excepting the positive real axis which constitutes the boundary of the domain corresponding to the boundary lines \({y}=0\) and \({y}=2 \pi\) . In other words the image of the above strip is the slit \(w\) -plane, the edge \(\phi=0\) corresponding to the boundary line \({y}=0\) and the edge \(\phi=2 \pi\) to the boundary line \(y=2 \pi\) . To represent the image of the next horizontal strip, \(2 \pi, we employ a second sheet, again split from \(0\) to \(\infty\) along the positive real axis and where \(\phi\) runs now from \(2 \pi\) to \(4 \pi\) . These two sheets are now to be joined along their edges \(\phi=2 \pi\) corresponding to the common boundary line \(y=2 \pi\) of the two strips in the \(z\) -plane. There remain now on this surface two free edges \(\phi=0\) and \(\phi=4 \pi\) ; to these further slit sheets are attached representing the images of the strips \(-2 \pi<{y}<0\) and \(4 \pi<{y}<6 \pi\) . By continuing this process we arrive at a surface consisting of infinitely many sheets with edges correlated in the order of adjoining horizontal strips of width \(2 \pi\) in the \({z}\) -plane. The surface thus obtained is the Riemann surface of the function \[{z}=\log {w},\] the inverse function of \(w=e^{z}\) . This surface which has branch points at \(0\) and \(\infty\) of an infinite order, permits us to define the logarithm, which is infinitely many valued, in terms of a one-to-one mapping. For we have \[z=x+i y=\log w\] and, consequently, \[x=\log \rho; \quad y=\phi\] where, in the simple \(w\) -plane, \(\phi\) is defined only to within an added multiple of \(2 \pi\) . The logarithm has infinitely many determinations \[\log w=\log |w|+i(\operatorname{am}(w) \pm 2 \pi n).\] On the Riemann surface, however, \(\operatorname{am}(w) \pm 2 \pi n=\phi\) is uniquely determined (the amplitudes of corresponding points in successive sheets differ by \(2 \pi\) ) and the function \(\log w\) is uniquely defined. We mention finally that the particular determination of the logarithm \[\log {w}=\log |{w}|+ i\operatorname{am}(w)\] where \(0 \leq \operatorname{am}(w) < 2\pi\) is called the principal value of the logarithm.

We should like to point out that if we let the point \(w\) describe a simple closed curve in the \(w\) -plane not enclosing the origin \(w=0\) then \(\log w\) returns to its initial value as \(w\) returns to its initial point on the curve, while if \(w\) describes a simple closed curve enclosing the origin then the value of \(\log w\) changes by \(\pm 2 \pi i\) according to whether the curve is described in the positive or negative sense of rotation.

With the use of the logarithm it is now possible to give a unique definition of the function \(w=z^{\alpha}\) for arbitrary complex \(\alpha\) . For simplicity we take \(\alpha\) real and positive. We set \[w=z^{\alpha}=e^{\log z^{\alpha}}=e^{\alpha \log z}\] where \(\log {z}\) is not considered to be in the \({z}\) -plane but in its Riemann surface where it is a single-valued function of position. Hence it follows that the function \(w=e^{\alpha \log z}\) is also a single-valued function on the Riemann surface of \(\log z\) . Furthermore, from the periodicity of the exponential function and from the fact that \(\log {z}\) changes its value by \(2 \pi i\) upon passage to corresponding points in successive sheets of the Riemann surface, it follows if \(\alpha\) is rational say \(\alpha=\frac{p}{q}\) , where \(p\) and \(q\) are relatively prime integers, that \(w\) returns to its initial value as \(\operatorname{am}(z)\) increases by \(2 \pi {q}\) . Thus, for a unique representation of \(w=z^{p/q}\) , it is sufficient to consider a Riemann surface, of only \(q\) sheets, in fact, the Riemann surface of \(\sqrt[q]{{z}}\) . Furthermore, this Riemann surface is not mapped on the simple \(w\) -plane in a one-to-one manner. Rather, as one sees readily by writing \(w=\left(z^{1 / q}\right)^{p}\) , this function maps the above Riemann surface on a Riemann surface over the \(w\) -plane belonging to the function \(\sqrt[p]{w}\) . It is between these two Riemann surfaces that the function \(w=z^{p / q}\) establishes a one-to-one and conformal correspondence.

As an application of the different functions and their mapping properties let us map the lens consisting of two circular arcs and including an angle \(\pi / \alpha\) in the \(z\) -plane onto the unit circle in the \(w\) -plane. To this end we apply a linear transformation to the \(z\) -plane mapping the point \(z=-1\) on the origin and the point \(z=+1\) on the point at infinity. This is done by \(w_{1}=\frac{1+z}{1-z}\) . Clearly the lens is mapped by this function onto a wedge of central angle \(\pi / \alpha\) at \(w_{1}=0\) in the \(w_{1}\) -plane. The passage from the wedge to the unit circle is accomplished simply by mapping the wedge first on the half-plane and then the half-plane on the unit circle. To carry this out we introduce the function \(w_{2}=w_{1}^{\alpha}=\big(\frac{1+z}{1-z}\big)^{\alpha}\) which maps the wedge on the right half-plane \(\operatorname{Re}(w_{2}) > 0\) . Upon rotating the \(w_{2}\) -plane by \(\frac{\pi}{2}\) the wedge is now mapped on the upper half-plane: \(w_{3}=e^{i \pi / 2} w_{2}\) . To this we now apply the transformation carrying the upper half-plane into the unit circle: \(w=e^{i \lambda} \frac{w_{3}-b}{w_{3}-\overline{b}}\) , where \(\operatorname{Im}(b) > 0\) . We may choose \(\lambda\) and \(b\) so that the point \(z=0\) is mapped on \(w=0\) and that \(\frac{d w}{d z}>0\) at that point. A simple calculation gives \(b=i\) and \(\lambda=\pi\) and \[w=\frac{(1-z)^{\alpha}-(1+z)^{\alpha}}{(1-z)^{\alpha}+(1+z)^{\alpha}}.\] 

2.4.4 The Function \(\boldsymbol{w=\frac{1}{2}\left(z+\frac{1}{z}\right)}\) 

The conformal mapping defined by this function presents special features of interest, and is worth studying in detail.

It is seen that \(\frac{d w}{d z}=0\) at the points \(z= \pm 1\) . The image points \(w= \pm 1\) are branch points, at which the mapping fails to be conformal. In polar coordinates \[u=\frac{1}{2}\left(r+\frac{1}{r}\right) \cos \theta; \quad v=\frac{1}{2}\left(r-\frac{1}{r}\right) \sin \theta\] for \(r= \textit{constant}\) , we obtain, eliminating \(\theta\) \[\tag{4.41} \frac{4 u^{2}}{\left(c+\frac{1}{c}\right)^{2}}+\frac{4 v^{2}}{\left(c-\frac{1}{c}\right)^{2}}=1\] which are a family of confocal ellipses in the \(w\) -plane, having foci at \(w \pm 1\) . The orthogonal family is obtained by setting \(\theta= \textit{constant}\) and eliminating \(r\) : \[\frac{u^{2}}{\cos ^{2} \theta}-\frac{v^{2}}{\sin ^{2} \theta}=1\] which are a family of confocal hyperbolas, all having the same foci, \(w= \pm 1\) . As \(r \rightarrow 1\) the corresponding ellipse (4.41) tends to the straight line joining \(w=+1\) and \(w=-1\) , covered twice. If we regard the line as a slit then the circumference of the unit circle is mapped on this slit. Every ellipse corresponds to two circles \(r=c\) and \(r=\frac{1}{c}\) , one of which is outside the unit circle, the other inside. To avoid this double valuedness we introduce another sheet over the \(w\) -plane joined to the first across the slit – upper edge to lower, lower to upper. The interior of the unit circle is mapped into one of the sheets, the exterior into the other sheet.

This function, therefore, solved the problem of mapping the exterior of the unit circle on a plane cut by a horizontal slit. Points lying opposite on the upper and lower edges of the slit are to be regarded as different boundary points.

Exercises

Exercise 2.7 . Map the exterior of the symmetric lens of the diagram in a one-to-one conformal manner onto the entire \(w\) -plane slit from \(u=-1\) to \(u=+1\) so that \({z}=\infty\) corresponds to \({w}=\infty\) and \({z}=1\) corresponds to \({w}=1\) .

Exercise 2.8 . Map conformally the interior of the unit circle \(|z|<1\) slit along the real axis from \(z=-1\) to \({z}=-{h}\) ; ( \(1>{h}>0\) ) onto a circle about the origin of the \(w\) -plane so that \({z}=0\) maps into \(w=0\) and \(\frac{d w}{d z}=1\) for \(z=0\) .