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Theorem 11. An analytic function in a region \( \Omega \) whose derivative vanishes identically must reduce to a constant. The same is true if either the real part, the imaginary part, the modulus, or the argument is constant. | The vanishing of the derivative implies that \( \partial u/\partial x,\partial u/\partial y,\partial v/\partial x \) , \( \partial v/\partial y \) are all zero. It follows that \( u \) and \( v \) are constant on any line segment in \( \Omega \) which is parallel to one of the coordinate axes. In Sec. 1.3 we remarked, ... | Yes |
Theorem 12. If \( {z}_{1},{z}_{2},{z}_{3},{z}_{4} \) are distinct points in the extended plane and \( T \) any linear transformation, then \( \left( {T{z}_{1}, T{z}_{2}, T{z}_{3}, T{z}_{4}}\right) = \left( {{z}_{1},{z}_{2},{z}_{3},{z}_{4}}\right) \) . | The proof is immediate, for if \( {Sz} = \left( {z,{z}_{2},{z}_{3},{z}_{4}}\right) \), then \( S{T}^{-1} \) carries \( T{z}_{2}, T{z}_{3}, T{z}_{4} \) into \( 1,0,\infty \) . By definition we have hence\n\n\[ \left( {T{z}_{1}, T{z}_{2}, T{z}_{3}, T{z}_{4}}\right) = S{T}^{-1}\left( {T{z}_{1}}\right) = S{z}_{1} = \left( ... | Yes |
Theorem 13. The cross ratio \( \left( {{z}_{1},{z}_{2},{z}_{3},{z}_{4}}\right) \) is real if and only if the four points lie on a circle or on a straight line. | This is evident by elementary geometry, for we obtain\n\n\[ \arg \left( {{z}_{1},{z}_{2},{z}_{3},{z}_{4}}\right) = \arg \frac{{z}_{1} - {z}_{3}}{{z}_{1} - {z}_{4}} - \arg \frac{{z}_{2} - {z}_{3}}{{z}_{2} - {z}_{4}} \]\n\nand if the points lie on a circle this difference of angles is either 0 or \( \pm \pi \) , dependin... | Yes |
Theorem 1. The line integral \( {\int }_{\gamma }{pdx} + {qdy} \), defined in \( \Omega \), depends only on the end points of \( \gamma \) if and only if there exists a function \( U\left( {x, y}\right) \) in \( \Omega \) with the partial derivatives \( \partial U/\partial x = p,\partial U/\partial y = q \) . | The sufficiency follows at once, for if the condition is fulfilled we can write, with the usual notations,\n\n\[ \n{\int }_{\gamma }{pdx} + {qdy} = {\int }_{a}^{b}\left( {\frac{\partial U}{\partial x}{x}^{\prime }\left( t\right) + \frac{\partial U}{\partial y}{y}^{\prime }\left( t\right) }\right) {dt} = {\int }_{a}^{b}... | Yes |
Theorem 3. Let \( f\left( z\right) \) be analytic on the set \( {R}^{\prime } \) obtained from a rectangle \( R \) by omitting a finite number of interior points \( {\zeta }_{j} \) . If it is true that\n\n\[ \mathop{\lim }\limits_{{z \rightarrow {\zeta }_{j}}}\left( {z - {\zeta }_{j}}\right) f\left( z\right) = 0 \]\n\n... | It is sufficient to consider the case of a single exceptional point \( \zeta \), for evidently \( R \) can be divided into smaller rectangles which contain at most one \( {\zeta }_{j} \) .\n\nWe divide \( R \) into nine rectangles, as shown in Fig. 4-3, and apply\n\n \) is analytic in an open disk \( \Delta \), then\n\n\[{\int }_{\gamma }f\left( z\right) {dz} = 0\]\n\nfor every closed curve \( \gamma \) in \( \Delta \) . | The proof is a repetition of the argument used in proving the second half of Theorem 1. We define a function \( F\left( z\right) \) by\n\n\[F\left( z\right) = {\int }_{\sigma }{fdz}\]\n\nwhere \( \sigma \) consists of the horizontal line segment from the center \( \left( {{x}_{0},{y}_{0}}\right) \) to \( \left( {x,{y}_... | Yes |
Theorem 5. Let \( f\left( z\right) \) be analytic in the region \( {\Delta }^{\prime } \) obtained by omitting a finite number of points \( {\zeta }_{j} \) from an open disk \( \Delta \) . If \( f\left( z\right) \) satisfies the condition \( \mathop{\lim }\limits_{{z \rightarrow {\zeta }_{j}}}\left( {z - {\zeta }_{j}}\... | The proof must be modified, for we cannot let \( \sigma \) pass through the exceptional points. Assume first that no \( {\zeta }_{j} \) lies on the lines \( x = {x}_{0} \) and \( y = {y}_{0} \) . It is then possible to avoid the exceptional points by letting \( \sigma \) consist of three segments (Fig. 4-4). By an obvi... | Yes |
Lemma 1. If the piecewise differentiable closed curve \( \gamma \) does not pass through the point \( a \), then the value of the integral\n\n\[ \n{\int }_{\gamma }\frac{dz}{z - a}\n\]\n\nis a multiple of \( {2\pi i} \) . | The simplest proof is computational. If the equation of \( \gamma \) is \( z = z\left( t\right) \) , \( \alpha \leqq t \leqq \beta \), let us consider the function\n\n\[ \nh\left( t\right) = {\int }_{\alpha }^{t}\frac{{z}^{\prime }\left( t\right) }{z\left( t\right) - a}{dt}\n\]\n\nIt is defined and continuous on the cl... | Yes |
Lemma 2. Let \( {z}_{1},{z}_{2} \) be two points on a closed curve \( \gamma \) which does not pass through the origin. Denote the subarc from \( {z}_{1} \) to \( {z}_{2} \) in the direction of the curve by \( {\gamma }_{1} \), and the subarc from \( {z}_{2} \) to \( {z}_{1} \) by \( {\gamma }_{2} \) . Suppose that \( ... | For the proof we draw the half lines \( {L}_{1} \) and \( {L}_{2} \) from the origin through \( {z}_{1} \) and \( {z}_{2} \) (Fig. 4-5). Let \( {\zeta }_{1},{\zeta }_{2} \) be the points in which \( {L}_{1},{L}_{2} \) intersect a circle \( C \) about the origin. If \( C \) is described in the positive sense, the arc \(... | Yes |
Theorem 6. Suppose that \( f\left( z\right) \) is analytic in an open disk \( \Delta \), and let \( \gamma \) be a closed curve in \( \Delta \) . For any point a not on \( \gamma \)\n\n\[ n\left( {\gamma, a}\right) \cdot f\left( a\right) = \frac{1}{2\pi i}{\int }_{\gamma }\frac{f\left( z\right) {dz}}{z - a} \]\n\nwhere... | In this statement we have suppressed the requirement that \( a \) be a point in \( \Delta \) . We have done so in view of the obvious interpretation of the formula (20) for the case that \( a \) is not in \( \Delta \) . Indeed, in this case \( n\left( {\gamma, a}\right) \) and the integral in the right-hand member are ... | Yes |
Lemma 3. Suppose that \( \varphi \left( \zeta \right) \) is continuous on the arc \( \gamma \) . Then the function\n\n\[ \n{F}_{n}\left( z\right) = {\int }_{\gamma }\frac{\varphi \left( \zeta \right) }{{\left( \zeta - z\right) }^{n}} \n\]\n\nis analytic in each of the regions determined by \( \gamma \), and its derivat... | We prove first that \( {F}_{1}\left( z\right) \) is continuous. Let \( {z}_{0} \) be a point not on \( \gamma \) , and choose the neighborhood \( \left| {z - {z}_{0}}\right| < \delta \) so that it does not meet \( \gamma \) . By restricting \( z \) to the smaller neighborhood \( \left| {z - {z}_{0}}\right| < \delta /2 ... | Yes |
Theorem 7. Suppose that \( f\left( z\right) \) is analytic in the region \( {\Omega }^{\prime } \) obtained by omitting a point a from a region \( \Omega \) . A necessary and sufficient condition that there exist an analytic function in \( \Omega \) which coincides with \( f\left( z\right) \) in \( {\Omega }^{\prime } ... | The necessity and the uniqueness are trivial since the extended function must be continuous at \( a \) . To prove the sufficiency we draw a circle \( C \) about \( a \) so that \( C \) and its inside are contained in \( \Omega \) . Cauchy’s formula is valid, and we can write\n\n\[ f\left( z\right) = \frac{1}{2\pi i}{\i... | Yes |
Theorem 8. If \( f\left( z\right) \) is analytic in a region \( \Omega \), containing \( a \), it is possible to write\n\n\[ f\left( z\right) = f\left( a\right) + \frac{{f}^{\prime }\left( a\right) }{1!}\left( {z - a}\right) + \frac{{f}^{\prime \prime }\left( a\right) }{2!}{\left( z - a}\right) }^{2} + \cdots \]\n\n\[ ... | This finite development must be well distinguished from the infinite Taylor series which we will study later. It is, however, the finite development (28) which is the most useful for the study of the local properties of \( f\left( z\right) \) . Its usefulness is enhanced by the fact that \( {f}_{n}\left( z\right) \) ha... | Yes |
Theorem 9. An analytic function comes arbitrarily close to any complex value in every neighborhood of an essential singularity. | If the assertion were not true, we could find a complex number \( A \) and \( {a\delta } > 0 \) such that \( \left| {f\left( z\right) - A}\right| > \delta \) in a neighborhood of \( a \) (except for \( z = a) \) . For any \( \alpha < 0 \) we have then \( \mathop{\lim }\limits_{{z \rightarrow a}}{\left| z - a\right| }^{... | Yes |
Theorem 10. Let \( {z}_{j} \) be the zeros of a function \( f\left( z\right) \) which is analytic in a disk \( \Delta \) and does not vanish identically, each zero being counted as many times as its order indicates. For every closed curve \( \gamma \) in \( \Delta \) which does not pass through a zero\n\n\[ \mathop{\su... | The function \( w = f\left( z\right) \) maps \( \gamma \) onto a closed curve \( \Gamma \) in the \( w \) -plane, and we find\n\n\[ {\int }_{\Gamma }\frac{dw}{w} = {\int }_{\gamma }\frac{{f}^{\prime }\left( z\right) }{f\left( z\right) }{dz} \]\n\nThe formula (32) has thus the following interpretation:\n\n\[ n\left( {\G... | Yes |
Theorem 11. Suppose that \( f\left( z\right) \) is analytic at \( {z}_{0}, f\left( {z}_{0}\right) = {w}_{0} \), and that \( f\left( z\right) - {w}_{0} \) has a zero of order \( n \) at \( {z}_{0} \) . If \( \varepsilon > 0 \) is sufficiently small, there exists a corresponding \( \delta > 0 \) such that for all a with ... | We can choose \( \varepsilon \) so that \( f\left( z\right) \) is defined and analytic for \( \left| {z - {z}_{0}}\right| \leqq \varepsilon \) and so that \( {z}_{0} \) is the only zero of \( f\left( z\right) - {w}_{0} \) in this disk. Let \( \gamma \) be the circle \( \left| {z - {z}_{0}}\right| = \varepsilon \) and \... | Yes |
Corollary 1. A nonconstant analytic function maps open sets onto open sets. | This is merely another way of saying that the image of every sufficiently small disk \( \left| {z - {z}_{0}}\right| < \varepsilon \) contains a neighborhood \( \left| {w - {w}_{0}}\right| < \delta \) . | No |
Corollary 2. If \( f\left( z\right) \) is analytic at \( {z}_{0} \) with \( {f}^{\prime }\left( {z}_{0}\right) \neq 0 \), it maps a neighborhood of \( {z}_{0} \) conformally and topologically onto a region. | From the continuity of the inverse function it follows in the usual way that the inverse function is analytic, and hence the inverse mapping is likewise conformal. Conversely, if the local mapping is one to one, Theorem 11 can hold only with \( n = 1 \), and hence \( {f}^{\prime }\left( {z}_{0}\right) \) must be differ... | Yes |
Theorem 12. (The maximum principle.) If \( f\left( z\right) \) is analytic and nonconstant in a region \( \Omega \), then its absolute value \( \left| {f\left( z\right) }\right| \) has no maximum in \( \Omega \) . | The proof is clear. If \( {w}_{0} = f\left( {z}_{0}\right) \) is any value taken in \( \Omega \), there exists a neighborhood \( \left| {w - {w}_{0}}\right| < \varepsilon \) contained in the image of \( \Omega \). In this neighborhood there are points of modulus \( > \left| {w}_{0}\right| \), and hence \( \left| {f\lef... | Yes |
Theorem 13. If \( f\left( z\right) \) is analytic for \( \left| z\right| < 1 \) and satisfies the conditions \( \left| {f\left( z\right) }\right| \leqq 1, f\left( 0\right) = 0 \), then \( \left| {f\left( z\right) }\right| \leqq \left| z\right| \) and \( \left| {{f}^{\prime }\left( 0\right) }\right| \leqq 1 \) . If \( \... | We apply the maximum principle to the function \( {f}_{1}\left( z\right) \) which is equal to \( f\left( z\right) /z \) for \( z \neq 0 \) and to \( {f}^{\prime }\left( 0\right) \) for \( z = 0 \) . On the circle \( \left| z\right| = r < 1 \) it is of absolute value \( \leqq 1/r \), and hence \( \left| {{f}_{1}\left( z... | Yes |
Theorem 14. A region \( \Omega \) is simply connected if and only if \( n\left( {\gamma, a}\right) = 0 \) for all cycles \( \gamma \) in \( \Omega \) and all points a which do not belong to \( \Omega \) . | This alternative condition is also very suggestive. It states that a closed curve in a simply connected region cannot wind around any point which does not belong to the region. It seems quite evident that this condition is not fulfilled in the case of a region with a hole.\n\nThe necessity of the condition is almost tr... | Yes |
Corollary 1. If \( f\left( z\right) \) is analytic in a simply connected region \( \Omega \), then (40) holds for all cycles \( \gamma \) in \( \Omega \) . | Before proving the theorem, we make an observation which ties up with the considerations in Section 1.3. As pointed out in that connection, the validity of (40) for all closed curves \( \gamma \) in a region means that the line integral of \( {fdz} \) is independent of the path, or that \( {fdz} \) is an exact differen... | No |
Corollary 2. If \( f\left( z\right) \) is analytic and \( \neq 0 \) in a simply connected region \( \Omega \) , then it is possible to define single-valued analytic branches of \( \log f\left( z\right) \) and \( \sqrt[n]{f\left( z\right) } \) in \( \Omega \) . | In fact, we know that there exists an analytic function \( F\left( z\right) \) in \( \Omega \) such that \( {F}^{\prime }\left( z\right) = {f}^{\prime }\left( z\right) /f\left( z\right) \) . The function \( f\left( z\right) {e}^{-F\left( z\right) } \) has the derivative zero and is therefore a constant. Choosing a poin... | Yes |
Theorem 17. Let \( f\left( z\right) \) be analytic except for isolated singularities \( {a}_{j} \) in a region \( \Omega \) . Then\n\n\[ \frac{1}{2\pi i}{\int }_{\gamma }f\left( z\right) {dz} = \mathop{\sum }\limits_{j}n\left( {\gamma ,{a}_{j}}\right) {\operatorname{Res}}_{z = {a}_{j}}f\left( z\right) \]\n\nfor any cyc... | In the applications it is frequently the case that each \( n\left( {\gamma ,{a}_{\imath }}\right) \) is either 0 or 1. Then we have simply\n\n\[ \frac{1}{2\pi i}{\int }_{\gamma }f\left( z\right) {dz} = \mathop{\sum }\limits_{j}{\operatorname{Res}}_{z = {a}_{j}}f\left( z\right) \]\n\nwhere the sum is extended over all s... | No |
Theorem 19. If \( {u}_{1} \) and \( {u}_{2} \) are harmonic in a region \( \Omega \), then\n\n\[{\int }_{\gamma }{u}_{1} * d{u}_{2} - {u}_{2} * d{u}_{1} = 0\]\nfor every cycle \( \gamma \) which is homologous to zero in \( \Omega \) . | For \( {u}_{1} = 1,{u}_{2} = u \) the formula reduces to (58). In the classical notation (60) would be written as\n\n\[{\int }_{\gamma }\left( {{u}_{1}\frac{\partial {u}_{2}}{\partial n} - {u}_{2}\frac{\partial {u}_{1}}{\partial n}}\right) \left| {dz}\right| = 0.\] | No |
Theorem 20. The arithmetic mean of a harmonic function over concentric circles \( \left| z\right| = r \) is a linear function of \( \log r \) , | \n\[\n\frac{1}{2\pi }{\int }_{\left| z\right| = r}{ud\theta } = \alpha \log r + \beta\n\]\n\nand if \( u \) is harmonic in a disk \( \alpha = 0 \) and the arithmetic mean is constant.\n\nIn the latter case \( \beta = u\left( 0\right) \), by continuity, and changing to a new origin we find\n\n\[\nu\left( {z}_{0}\right) ... | Yes |
Theorem 23. The function \( {P}_{U}\left( z\right) \) is harmonic for \( \left| z\right| < 1 \), and\n\n\[ \mathop{\lim }\limits_{{z \rightarrow {e}^{i{\theta }_{0}}}}{P}_{U}\left( z\right) = U\left( {\theta }_{0}\right) \]\n\nprovided that \( U \) is continuous at \( {\theta }_{0} \) . | We have already remarked that \( {P}_{U} \) is harmonic. To study the boundary behavior, let \( {C}_{1} \) and \( {C}_{2} \) be complementary arcs of the unit circle, and denote by \( {U}_{1} \) the function which coincides with \( U \) on \( {C}_{1} \) and vanishes on \( {C}_{2} \), by \( {U}_{2} \) the corresponding ... | Yes |
Theorem 24. Let \( {\Omega }^{ + } \) be the part in the upper half plane of a symmetric region \( \Omega \), and let \( \sigma \) be the part of the real axis in \( \Omega \) . Suppose that \( v\left( x\right) \) is continuous in \( {\Omega }^{ + } \cup \sigma \), harmonic in \( {\Omega }^{ + } \), and zero on \( \sig... | For the proof we construct the function \( V\left( z\right) \) which is equal to \( v\left( z\right) \) in \( {\Omega }^{ + },0 \) on \( \sigma \), and equal to \( - v\left( \bar{z}\right) \) in the mirror image of \( {\Omega }^{ + } \) . We have to show that \( V \) is harmonic on \( \sigma \) . For a point \( {x}_{0}... | Yes |
Theorem 1. Suppose that \( {f}_{n}\left( z\right) \) is analytic in the region \( {\Omega }_{n} \), and that the sequence \( \left\{ {{f}_{n}\left( z\right) }\right\} \) converges to a limit function \( f\left( z\right) \) in a region \( \Omega \), uniformly on every compact subset of \( \Omega \) . Then \( f\left( z\r... | The analyticity of \( f\left( z\right) \) follows most easily by use of Morera’s theorem (Chap. 4, Sec. 2.3). Let \( \left| {z - a}\right| \leqq r \) be a closed disk contained in \( \Omega \) ; the assumption implies that this disk lies in \( {\Omega }_{n} \) for all \( n \) greater than a certain \( {n}_{0} \cdot \da... | Yes |
Theorem 2. If the functions \( {f}_{n}\left( z\right) \) are analytic and \( \neq 0 \) in a region \( \Omega \) , and if \( {f}_{n}\left( z\right) \) converges to \( f\left( z\right) \), uniformly on every compact subset of \( \Omega \), then \( f\left( z\right) \) is either identically zero or never equal to zero in \... | Suppose that \( f\left( z\right) \) is not identically zero. The zeros of \( f\left( z\right) \) are in any case isolated. For any point \( {z}_{0} \in \Omega \) there is therefore a number \( r > 0 \) such that \( f\left( z\right) \) is defined and \( \neq 0 \) for \( 0 < \left| {z - {z}_{0}}\right| \leqq r \) . In pa... | Yes |
Theorem 3. If \( f\left( z\right) \) is analytic in the region \( \Omega \), containing \( {z}_{0} \), then the representation\n\n\[ f\left( z\right) = f\left( {z}_{0}\right) + \frac{{f}^{\prime }\left( {z}_{0}\right) }{1!}\left( {z - {z}_{0}}\right) + \cdots + \frac{{f}^{\left( n\right) }\left( {z}_{0}\right) }{n!}{\l... | The radius of convergence of the Taylor series is thus at least equal to the shortest distance from \( {z}_{0} \) to the boundary of \( \Omega \) . It may well be larger, but if it is there is no guarantee that the series still represents \( f\left( z\right) \) at all points which are simultaneously in \( \Omega \) and... | Yes |
Theorem 4. Let \( \left\{ {b}_{\nu }\right\} \) be a sequence of complex numbers with \( \mathop{\lim }\limits_{{\nu \rightarrow \infty }}{b}_{\nu } = \infty \) , and let \( {P}_{\nu }\left( \zeta \right) \) be polynomials without constant term. Then there are functions which are meromorphic in the whole plane with pol... | We may suppose that no \( {b}_{\nu } \) is zero. The function \( {P}_{\nu }\left( {1/\left( {z - {b}_{\nu }}\right) }\right) \) is analytic for \( \left| z\right| < \left| {b}_{\nu }\right| \) and can thus be expanded in a Taylor series about the origin. We choose for \( {p}_{\nu }\left( z\right) \) a partial sum of th... | Yes |
Theorem 5. The infinite product \( \mathop{\prod }\limits_{1}^{\infty }\left( {1 + {a}_{n}}\right) \) with \( 1 + {a}_{n} \neq 0 \) converges simultaneously with the series \( \mathop{\sum }\limits_{1}^{\infty }\log \left( {1 + {a}_{n}}\right) \) whose terms represent the values of the principal branch of the logarithm... | The question of convergence of a product can thus be reduced to the more familiar question concerning the convergence of a series. It can be further reduced by observing that the series (16) converges absolutely at the same time as the simpler series \( \sum \left| {a}_{n}\right| \) . This is an immediate consequence o... | Yes |
Theorem 9. For \( \sigma = \operatorname{Re}s > 1 \) ,\n\n\[ \frac{1}{\zeta \left( s\right) } = \mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 - {p}_{n}^{-s}}\right) . \] | According to Theorem 6 the infinite product converges uniformly for \( \sigma \geqq {\sigma }_{0} > 1 \) if the same is true of the series \( \mathop{\sum }\limits_{1}^{\infty }\left| {p}_{n}^{-s}\right| = \mathop{\sum }\limits_{1}^{\infty }{p}_{n}^{-\sigma } \) . Since the latter is obtained by omitting terms of \( \m... | Yes |
Theorem 10. For \( \sigma > 1 \) , \[ \zeta \left( s\right) = - \frac{\Gamma \left( {1 - s}\right) }{2\pi i}{\int }_{C}\frac{{\left( -z\right) }^{s - 1}}{{e}^{z} - 1}{dz} \] where \( {\left( -z\right) }^{s - 1} \) is defined on the complement of the positive real axis as \( {e}^{\left( {\bullet - 1}\right) \log \left( ... | The integral is obviously convergent. By Cauchy's theorem its value does not depend on the shape of \( C \) as long as \( C \) does not enclose any multiples of \( {2\pi i} \) . In particular, we are free to let \( r \) tend to zero. It is readily seen that the integral over the circle tends to zero with \( r \) . In t... | Yes |
\[ \zeta \left( s\right) = {2}^{s}{\pi }^{s - 1}\sin \frac{\pi s}{2}\Gamma \left( {1 - s}\right) \zeta \left( {1 - s}\right) . \] | For the proof we make use of the path \( {C}_{n} \) in Fig. 5-1; we assume that the square part lies on the lines \( t = \pm \left( {{2n} + 1}\right) \pi \) and \( \sigma = \pm \left( {{2n} + 1}\right) \pi \) . The cycle \( {C}_{n} - C \) has winding number one about the points \( \pm {2m\pi i} \) with \( m = 1,\ldots,... | Yes |
Theorem 12. A family \( \mathfrak{F} \) is normal if and only if its closure \( {\mathfrak{F}}^{ - } \) with respect to the distance function (65) is compact. | It is also customary to say that \( \mathfrak{F} \) is relatively compact if \( {\mathfrak{F}}^{ - } \) is compact. Thus, normal and relatively compact families are the same. We shall now relate the notion of normal families to total boundedness. If \( \mathfrak{F} \) is normal, then \( {\mathfrak{F}}^{ - } \) is compa... | Yes |
Theorem 13. The family \( \mathfrak{F} \) is totally bounded if and only if to every compact set \( E \subset \Omega \) and every \( \varepsilon > 0 \) it is possible to find \( {f}_{1},\ldots ,{f}_{n}\epsilon \mathfrak{F} \) such that every \( f \in \mathfrak{F} \) satisfies \( d\left( {f,{f}_{j}}\right) < \varepsilon... | If \( \mathfrak{F} \) is totally bounded there exist \( {f}_{1},\ldots ,{f}_{n} \) such that, for any \( f \in \mathfrak{F} \) , \( \rho \left( {f,{f}_{j}}\right) < \varepsilon \) for some \( {f}_{j}.\; \) By (65) this implies \( {\delta }_{k}\left( {f,{f}_{j}}\right) < {2}^{k}\varepsilon \), or \( \delta \left( {f,{f}... | Yes |
Theorem 14. A family \( \mathfrak{F} \) of continuous functions with values in a metric space \( S \) is normal in the region \( \Omega \) of the complex plane if and only if\n\n(i) & is equicontinuous on every compact set \( E \subset \Omega \) ;\n\n(ii) for any \( z \in \Omega \) the values \( f\left( z\right), f \in... | We give two proofs of the necessity of (i). Assume that \( \mathfrak{F} \) is normal and determine \( {f}_{1},\ldots ,{f}_{n} \) as in Theorem 13. Because each of these functions is uniformly continuous on \( E \) we can find a \( \delta > 0 \) such that \( d\left( {{f}_{j}\left( z\right) ,{f}_{j}\left( {z}_{0}\right) ... | Yes |
Theorem 15. A family \( \mathfrak{F} \) of analytic functions is normal with respect to \( \mathbf{C} \) if and only if the functions in \( \mathfrak{F} \) are uniformly bounded on every compact set. | To prove the sufficiency we prove equicontinuity. Let \( C \) be the boundary of a closed disk in \( \Omega \), of radius \( r \) . If \( z,{z}_{0} \) are inside \( C \) we obtain by Cauchy's integral theorem\n\n\[ f\left( z\right) - f\left( {z}_{0}\right) = \frac{1}{2\pi i}{\int }_{C}\left( {\frac{1}{\zeta - z} - \fra... | Yes |
Theorem 16. A locally bounded family of analytic functions has locally bounded derivatives. | This follows at once by the Cauchy representation of the derivative. If \( C \) is the boundary of a closed disk in \( \Omega \), of radius \( r \), then\n\n\[ \n{f}^{\prime }\left( z\right) = \frac{1}{2\pi i}{\int }_{C}\frac{f\left( \zeta \right) {d\zeta }}{{\left( \zeta - z\right) }^{2}}. \n\]\n\nHence \( \left| {{f}... | Yes |
Theorem 17. A family of analytic or meromorphic functions \( f \) is normal in the classical sense if and only if the expressions\n\n(58)\n\n\[ \rho \left( f\right) = \frac{2\left| {{f}^{\prime }\left( z\right) }\right| }{1 + {\left| f\left( z\right) \right| }^{2}} \] \n\nare locally bounded. | The geometric meaning of the quantity \( \rho \left( f\right) \) is rather evident. Indeed, by use of the formula in Chap. 1, Sec. 2.4\n\n\[ d\left( {f\left( {z}_{1}\right), f\left( {z}_{2}\right) }\right) = \frac{2\left| {f\left( {z}_{1}\right) - f\left( {z}_{2}\right) }\right| }{{\left\lbrack \left( 1 + {\left| f\lef... | Yes |
Theorem 1. Given any simply connected region \( \Omega \) which is not the whole plane, and a point \( {z}_{0} \in \Omega \), there exists a unique analytic function \( f\left( z\right) \) in \( \Omega \) , normalized by the conditions \( f\left( {z}_{0}\right) = 0,{f}^{\prime }\left( {z}_{0}\right) > 0 \), such that \... | The uniqueness is easily proved, for if \( {f}_{1} \) and \( {f}_{2} \) are two such functions, then \( {f}_{1}\left\lbrack {{f}_{2}^{-1}\left( w\right) }\right\rbrack \) defines a one-to-one mapping of \( \left| w\right| < 1 \) onto itself. We know that such a mapping is given by a linear transformation \( S \) (Chap.... | Yes |
Theorem 2. Let \( f \) be a topological mapping of a region \( \Omega \) onto a region \( {\Omega }^{\prime } \) . If \( \left\{ {z}_{n}\right\} \) or \( z\left( t\right) \) tends to the boundary of \( \Omega \), then \( \left\{ {f\left( {z}_{n}\right) }\right\} \) or \( f\left( {z\left( t\right) }\right) \) tends to t... | Indeed, let \( K \) be a compact set in \( {\Omega }^{\prime } \) . Then \( {f}^{-1}\left( K\right) \) is a compact set in \( \Omega \), and there exists \( {n}_{0} \) (or \( {t}_{0} \) ) such that \( {z}_{n} \) (or \( z\left( t\right) \) ) is not in \( {f}^{-1}\left( K\right) \) for \( n > {n}_{0} \) (or \( t > {t}_{0... | Yes |
Theorem 3. Suppose that the boundary of a simply connected region \( \Omega \) contains a line segment \( \gamma \) as a one-sided free boundary arc. Then the function \( f\left( z\right) \) which maps \( \Omega \) onto the unit disk can be extended to a function which is analytic and one to one on \( \Omega \cup \gamm... | For the proof we consider a disk around \( {x}_{0}{\epsilon \gamma } \) which is so small that the half disk in \( \Omega \) does not contain the point \( {z}_{0} \) with \( f\left( {z}_{0}\right) = 0 \) . Then \( \log f\left( z\right) \) has a single-valued branch in the half disk, and its real part tends to 0 as \( z... | Yes |
Theorem 4. If the boundary of \( \Omega \) contains a free one-sided analytic arc \( \gamma \), then the mapping function has an analytic extension to \( \Omega \cup \gamma \), and \( \gamma \) is mapped on an arc of the unit circle. | We trust the reader to make the last statement more precise and to complete the proof. | No |
Theorem 6. A continuous function \( u\left( z\right) \) which satisfies condition (8) is necessarily harmonic. | Again, the condition need be satisfied only for sufficiently small \( r \) . If \( u \) satisfies (8), so does the difference between \( u \) and any harmonic function. Suppose that the disk \( \left| {z - {z}_{0}}\right| \leqq \rho \) is contained in \( \Omega \), the region where \( u \) is defined. By use of Poisson... | Yes |
Theorem 7. Consider a sequence of functions \( {u}_{n}\left( z\right) \), each defined and harmonic in a certain region \( {\Omega }_{n} \) . Let \( \Omega \) be a region such that every point in \( \Omega \) has a neighborhood contained in all but a finite number of the \( {\Omega }_{n} \), and assume moreover that in... | For the proof, suppose first that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{u}_{n}\left( {z}_{0}\right) = \infty \) for at least one point \( {z}_{0} \in \Omega \) . By assumption there exist \( r \) and \( m \) such that the functions \( {u}_{n}\left( z\right) \) are harmonic and form a nondecreasing sequence... | Yes |
Theorem 8. A continuous function \( v\left( z\right) \) is subharmonic in \( \Omega \) if and only if it satisfies the inequality\n\n(12)\n\n\[ v\left( {z}_{0}\right) \leqq \frac{1}{2\pi }{\int }_{0}^{2\pi }v\left( {{z}_{0} + r{e}^{i\theta }}\right) {d\theta } \]\n\nfor every disk \( \left| {z - {z}_{0}}\right| \leqq r... | The sufficiency follows by the fact that (12), rather than the mean-value property, is what is actually needed in order to show that \( v \) cannot have a maximum without being constant. Since \( v - u \) satisfies the same inequality, it follows that \( v \) is subharmonic.\n\nIn order to prove the necessity we form t... | Yes |
Lemma 2. Suppose that there exists a harmonic function \( \omega \left( z\right) \) in \( \Omega \) whose continuous boundary values \( \omega \left( \zeta \right) \) are strictly positive except at one point \( {\zeta }_{0} \) where \( \omega \left( {\zeta }_{0}\right) = 0 \) . Then, if \( f\left( \zeta \right) \) is ... | The lemma will be proved if we show that \( \mathop{\lim }\limits_{{z \rightarrow {\zeta }_{0}}}u\left( z\right) \leqq f\left( {\zeta }_{0}\right) + \varepsilon \) and \( \mathop{\lim }\limits_{{z \rightarrow {\zeta }_{0}}}u\left( z\right) \geqq f\left( {\zeta }_{0}\right) - \varepsilon \) for all \( \varepsilon > 0 \)... | Yes |
Theorem 10. The function \( F\left( z\right) \) effects a one-to-one conformal mapping of \( \Omega \) onto the annulus \( 1 < \left| w\right| < {e}^{{\lambda }_{1}} \) minus \( n - 2 \) concentric arcs situated on the circles \( \left| w\right| = {e}^{{\lambda }_{i}}, i = 2,\ldots, n - 1 \) . | The proof is by use of the argument principle. We know that \( F\left( z\right) \) is analytic with a constant modulus on each contour. The number of roots of the equation \( F\left( z\right) = {w}_{0} \) is given by\n\n(15)\n\n\[ \frac{1}{2\pi i}{\int }_{{C}_{1}}\frac{{F}^{\prime }\left( z\right) {dz}}{F\left( z\right... | Yes |
Lemma 3. The period \( {P}_{k}\left( {z}_{0}\right) \) equals the harmonic measure \( {\omega }_{k}\left( {z}_{0}\right) \) multiplied by \( {2\pi } \) . | The proof is another application of Theorem 21, Chap. 4. We express the fact that the integral of \( {\omega }_{k} * {dg} - g * d{\omega }_{k} \) over \( C - c \) must vanish. The integral over \( C \) reduces to \( {P}_{k}\left( {z}_{0}\right) \), and by the same computation as above the integral over \( c \) equals \... | Yes |
Theorem 11. The mappings determined by \( p\left( z\right) \) and \( q\left( z\right) \) are one to one, and the image of \( \Omega \) is a slit region whose complement consists of \( n \) vertical or horizontal segments, respectively (Fig. 6-5a, b). | The proof is quite similar to that of Theorem 10. This time the expression\n\n(17)\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{n}\frac{1}{2\pi i}{\int }_{{C}_{k}}\frac{{p}^{\prime }\left( z\right) {dz}}{p\left( z\right) - {w}_{0}} \]\n\nrepresents the number of zeros of \( p\left( z\right) - {w}_{0} \) minus the number of p... | Yes |
Theorem 1. A discrete module consists either of zero alone, of the integral multiples \( {n\omega } \) of a single complex number \( \omega \neq 0 \), or of all linear combinations \( {n}_{1}\omega + {n}_{2}{\omega }_{2} \) with integral coefficients of two numbers \( {\omega }_{1},{\omega }_{2} \) with nonreal ratio \... | As soon as \( M \) contains a number \( \omega \neq 0 \) it also contains one, call it \( {\omega }_{1} \), whose absolute value is a minimum. Indeed, if \( r \) is large enough the disk \( \left| z\right| \leqq r \) contains a point from \( M \), other than 0 . Because the points are isolated there are only a finite n... | Yes |
Theorem 2. There exists a basis \( \left( {{\omega }_{1},{\omega }_{2}}\right) \) such that the ratio \( \tau = {\omega }_{2}/{\omega }_{1} \) satisfies the following conditions: (i) Im \( \tau > 0 \) ,(ii) \( - \frac{1}{2} < \operatorname{Re}\tau \leqq \frac{1}{2} \) , (iii) \( \left| \tau \right| \geqq 1 \) ,(iv) Re ... | Proof. If we select \( {\omega }_{1} \) and \( {\omega }_{2} \) as in the proof of Theorem 1, then \( \left| {\omega }_{1}\right| \leqq \) \( \left| {\omega }_{2}\right| ,\left| {\omega }_{2}\right| \leqq \left| {{\omega }_{1} + {\omega }_{2}}\right| \), and \( \left| {\omega }_{2}\right| \leqq \left| {{\omega }_{1} - ... | Yes |
Theorem 4. The sum of the residues of an elliptic function is zero. | We may choose \( a \) so that none of the poles fall on the boundary of \( {P}_{a} \) . If the boundary \( \partial {P}_{a} \) is traced in the positive sense, the sum of the residues at the poles in \( {P}_{a} \) equals\n\n\[ \frac{1}{2\pi i}{\int }_{\partial {P}_{a}}f\left( z\right) {dz} \]\n\nBecause \( f \) has per... | Yes |
Theorem 5. A nonconstant elliptic function has equally many poles as it has zeros. | The poles and zeros of \( f \) are simple poles of \( {f}^{\prime }/f \), which is itself an elliptic function. The multiplicities are the residues of \( {f}^{\prime }/f \), counted positive for zeros and negative for poles. The theorem now follows from Theorem 4. | Yes |
Theorem 6. The zeros \( {a}_{1},\ldots ,{a}_{n} \) and poles \( {b}_{1},\ldots ,{b}_{n} \) of an elliptic function satisfy \( {a}_{1} + \cdots + {a}_{n} \equiv {b}_{1} + \cdots + {b}_{n}\left( {\;\operatorname{mod}\;M}\right) \) . | This is proved by considering the integral\n\n(8)\n\n\[ \frac{1}{2\pi i}{\int }_{\partial {P}_{a}}\frac{z{f}^{\prime }\left( z\right) }{f\left( z\right) }{dz} \]\n\nwhere we may again assume that there are no zeros or poles on the boundary. By the calculus of residues the integral equals \( {a}_{1} + \cdots \) + \( {a}... | Yes |
Theorem 7. The modular function \( \lambda \left( \tau \right) \) effects a one-to-one conformal mapping of the region \( \Omega \) onto the upper half plane. The mapping extends continuously to the boundary in such a way that \( \tau = 0,1,\infty \) correspond to \( \lambda = 1,\infty ,0 \) . | By reflection the region \( {\Omega }^{\prime } \) that is symmetric to \( \Omega \) with respect to the imaginary axis is mapped onto the lower half plane, and thus both regions together correspond to the whole plane, except for the points 0 and 1 . | Yes |
Theorem 8. Every point \( \tau \) in the upper half plane is equivalent under the congruence subgroup mod 2 to exactly one point in \( \bar{\Omega } \cup {\Omega }^{\prime } \) . | We refer to Fig. 7-4. The reader is asked to verify that the region \( \Delta \) is mapped on the shaded regions in the figure by means of the linear transformations \( \tau , - 1/\tau ,\tau - 1,1/\left( {1 - \tau }\right) ,\left( {\tau - 1}\right) /\tau ,\tau /\left( {1 - \tau }\right) \) which we shall denote by \( {... | No |
Theorem 1. Two analytic continuations \( {\bar{\gamma }}_{1} \) and \( {\bar{\gamma }}_{2} \) of a global analytic function \( \mathbf{f} \) along the same arc \( \gamma \) are either identical, or \( {\bar{\gamma }}_{1}\left( t\right) \neq {\bar{\gamma }}_{2}\left( t\right) \) for all \( t \) . | The proof is a triviality. Because \( \pi \) is a local homeomorphism the image of \( {\bar{\gamma }}_{1} - {\bar{\gamma }}_{2} \) cannot contain a point of the zero section without being contained in it. | No |
Theorem 3. If \( P\left( {w, z}\right) \) and \( Q\left( {w, z}\right) \) are relatively prime polynomials, there are only a finite number of values \( {z}_{0} \) for which the equations \( P\left( {w,{z}_{0}}\right) = 0 \) and \( Q\left( {w,{z}_{0}}\right) = 0 \) have a common root. | We suppose that \( P \) and \( Q \) are ordered according to decreasing powers of \( w \) and set \( Q\left( {w, z}\right) = {b}_{0}\left( z\right) {w}^{m} + \cdots + {b}_{m}\left( z\right) \) where \( {b}_{0}\left( z\right) \) is not identically zero. If \( P \) is divided by \( Q \), the division algorithm yields a q... | Yes |
Lemma 1. There exists an open disk \( \Delta \), containing \( {z}_{0} \), and \( n \) function elements \( \left( {{f}_{1},\Delta }\right) ,\left( {{f}_{2},\Delta }\right) ,\ldots ,\left( {{f}_{n},\Delta }\right) \) with these properties:\n\n(a) \( P\left( {{f}_{i}\left( z\right), z}\right) = 0 \) in \( \Delta \) ;\n\... | The polynomial \( P\left( {w,{z}_{0}}\right) \) has simple zeros at \( w = {w}_{i} \) . We determine \( \varepsilon > 0 \) so that the disks \( \left| {w - {w}_{i}}\right| \leqq \varepsilon \) do not overlap and denote the circles \( \left| {w - {w}_{i}}\right| = \varepsilon \) by \( {C}_{i} \) . Then \( P\left( {w,{z}... | Yes |
An analytic function is an algebraic function if it has a finite number of branches and at most algebraic singularities. | Every algebraic function \( w = \mathbf{f}\left( z\right) \) satisfies an irreducible equation \( P\left( {w, z}\right) = 0 \), unique up to a constant factor, and every such equation determines a corresponding algebraic function uniquely. | Yes |
Proposition 2.7. Complements are unique in a Boolean algebra. | Proof. Suppose for some some element \( a \) we had two complements \( x \) and \( y \) . Then\n\n\[ x = x \land \left( {x \vee a}\right) \]\n\n\[ = x \land 1 \]\n\n\[ = x \land \left( {y \vee a}\right) \]\n\n\[ = \left( {x \land y}\right) \vee \left( {x \land a}\right) \text{(since Boolean algebras are distributive)} ... | Yes |
Proposition 2.8. The following statements hold in Boolean algebras.\n\n(1) Meets and joins are idempotent: \( a \land a = a \), and \( a \vee a = a \) . | Proof. For each of these statments, we need prove only half; the other half follows dually by exchanging \( \vee \) with \( \land ,0 \) with \( 1 \), and \( \leq \) with \( \geq \) . (1) and (2) follow immediately from the definitions of meet and join. | No |
Theorem 2.12. In a Boolean algebra \( A \), an ideal \( I \subset A \) is maximal iff for every \( a \in A \), either \( a \in I \) or \( \neg a \in I \), but not both. | Proof. First, let \( {a}_{0} \) be an element of \( A \) and let \( I \) be an ideal of \( A \) containing neither \( {a}_{0} \) nor \( \neg {a}_{0} \) . We will show that \( I \) is not maximal. To do so, consider the set \( J \) of all elements of the form \( a \vee b \), where \( a \leq {a}_{0} \) and \( b \in I \) ... | Yes |
Lemma 2.13 (Maximal Ideal Theorem). Every proper ideal in a Boolean algebra is contained in some maximal ideal. | Proof. This proof follows Halmos and Givant [2], page 72.\n\nLet \( B \) be a Boolean algebra with some proper ideal \( I \) . Further assume \( B \) is countable. (The proof for uncountable Boolean algebras is similar, but requires the axiom of choice.) We may then enumerate the elements in \( B : {p}_{0},{p}_{1},{p}_... | Yes |
Lemma 2.16 (Homomorphism Theorem). Every proper ideal is the kernel of some epimorphism between Boolean algebras. | Proof. This proof follows Johnstone [3], Lemma I 2.1.\n\nLet \( A \) be a Boolean algebra, let \( I \) be any proper ideal of \( A \), and define a relation \( { \equiv }_{I} \) by \( a{ \equiv }_{I}b \) iff there exist \( i \) and \( j \) in \( I \) such that \( a \vee i = b \vee j \) . Then \( { \equiv }_{I} \) is an... | Yes |
Proposition 3.2. Recall that \( \mathcal{2} \) is the set \( \{ 0,1\} \) . Let \( A \) be an arbitrary nonempty set. Consider the set \( {\mathcal{2}}^{A} = \{ f : f \) is a function from \( A \) to \( \mathcal{2}\} \) . Then \( {\mathcal{2}}^{A} \) is a Stone space. | Proof. Endow 2 with the discrete topology. The set \( {\mathbf{2}}^{A} \) is homomorphic to the Cartesian product of 2 with itself, with one copy for each element of \( A \) ; give it the product topology. Tychonoff's Theorem guarantees that this set is compact and Hausdorff. To show that it is a Stone space, we need t... | Yes |
Lemma 3.3 (Existence Theorem). If \( p \) is a nonzero element of a Boolean algebra \( A \), then there exists a homomorphism \( f : A \rightarrow \mathbf{2} \) such that \( f\left( p\right) = 1 \) . | Proof. Let \( I \) be the principal ideal generated by \( \neg p \), as defined in Equation 2.14. Then by the Maximal Ideal Theorem (Lemma 2.13), there is some maximal ideal \( M \) containing \( \neg p \) . By Theorem 2.12, \( p \notin M \) . By Lemma 2.16, there exists some homomorphism \( f : A \rightarrow \mathbf{2... | Yes |
Proposition 3.4. Let \( A \) be a Boolean algebra, and consider the set \( \mathcal{S}\left( A\right) \subset {\mathcal{2}}^{A} \) of homomorphisms from \( A \) to 2. Then \( \mathcal{S}\left( A\right) \) is a Stone space. | Proof. Fix \( a \in A \) . For all \( x \in {\mathbf{2}}^{A},\left\{ {x}_{a}\right\} \) is open in \( \mathbf{2} \), since \( \mathbf{2} \) has the discrete topology, and sets of the form \( \left\{ {x \in {\mathbf{2}}^{a} : {x}_{a} = 1}\right\} \) and \( \left\{ {x \in {\mathbf{2}}^{a} : {x}_{a} = 0}\right\} \) are op... | Yes |
Lemma 3.8. If \( X \) is a Stone space and \( F \) is a separating field of clopen subsets of \( X \), then \( F \) is the dual algebra of \( X \) ; that is, it is the field of all clopen subsets of \( X \) . | Proof. We first show that every open set in \( X \) can be written as a union of finitely many sets of \( F \) . Since \( F \) separates points, it also separates points and closed sets. To prove this, first suppose \( C \) is a closed set and \( x \notin C \) is a point of \( X \) . Since \( X \) is Hausdorff, for eac... | Yes |
Theorem 3.9 (Stone Representation Theorem for Boolean Algebras). Every Boolean algebra is isomorphic to the dual algebra of its associated Stone space. | Proof. Let \( A \) be a Boolean algebra, and let \( B \) be the dual algebra of its Stone space. We need to find an isomorphism between \( A \) and \( B \) . Our culprit shall be the function \( f \), defined by \( f\left( p\right) = \{ x \in \mathcal{S}\left( A\right) : x\left( p\right) = 1\} \) . To convict \( f \) o... | Yes |
Lemma 1.1. For every convergence \( \xi \) and each \( \mathcal{G} \in \mathfrak{J}\left( \xi \right) \), one has\n\n\[ \n{\operatorname{adh}}_{\xi }\mathcal{G} = {\operatorname{adh}}_{J\xi }\mathcal{G} \n\] | Proof. Let \( x \in {\operatorname{adh}}_{J\xi }\mathcal{G} \) : there is a filter \( \mathcal{F}\# \mathcal{G} \) such that \( x \in \mathop{\lim }\limits_{{J\xi }}\mathcal{F} \) and thus by (1.7) for every \( \mathcal{H} \) in \( \mathfrak{J}\left( \xi \right) \) that meshes \( \mathcal{F} \), one has \( x \in {\oper... | Yes |
Theorem 1.2. Let \( f : X \rightarrow Y \) be a surjective map, \( \xi \) a convergence on \( X \) and \( \tau \) a convergence on \( Y \) . If \( J \) is a projection corresponding to \( \mathfrak{J}\left( \cdot \right) \) in (1.5), then \( f \) is a \( J \) -map if and only if (1.2) holds. | Proof. Suppose that \( \tau \geq J\left( {f\xi }\right) \) and let \( y \in {\operatorname{adh}}_{\tau }\mathcal{G} \) for \( \mathcal{G} \in \mathfrak{J}\left( {f\xi }\right) \) . Hence \( y \in {\operatorname{adh}}_{J\left( {f\xi }\right) }\mathcal{G} \) and by Lemma 1.1, \( y \in {\operatorname{adh}}_{f\xi }\mathcal... | Yes |
Proposition 3.1. If \( \xi \leq \theta \) implies \( \mathfrak{E}\left( \theta \right) \subset \mathfrak{E}\left( \xi \right) \), then the class of \( \mathfrak{E}\left( \cdot \right) \) -based (respectively \( \mathfrak{E}\left( \cdot \right) \) -founded) convergences is inf-closed. | Proof. Let \( \mathcal{D} \) be a set of \( \mathfrak{E}\left( \cdot \right) \) -based (respectively \( \mathfrak{E}\left( \cdot \right) \) -founded) convergences on \( Y \) and let \( y \in \mathop{\lim }\limits_{{\bigwedge \mathcal{D}}}\mathcal{F} \) . Then, there is \( \tau \in \mathcal{D} \) such that \( y \in \mat... | Yes |
If \( \xi \leq \theta \) implies \( \mathfrak{E}\left( \theta \right) \subset \mathfrak{E}\left( \xi \right) \) and besides for every \( \xi ,\mathfrak{E}\left( {\xi \vee \mathfrak{E}\left( \xi \right) }\right) = \) \( \mathfrak{E}\left( \xi \right) \) (respectively \( \mathfrak{E}\left( {\xi \mid \mathfrak{E}\left( \x... | Proof. Let \( \mathfrak{E}\left( \cdot \right) \) fulfil the assumptions and let \( \xi \) be a convergence. Since \( \mathfrak{E}\left( \xi \right) = \) \( \mathfrak{E}\left( {\xi \vee \mathfrak{E}\left( \xi \right) }\right) \) (respectively \( = \mathfrak{E}\left( {\xi \mid \mathfrak{E}\left( \xi \right) }\right) \) ... | Yes |
Proposition 3.3. If \( \mathbb{E} \) is one of the classes (3.3), then for every map \( f \) , \[ \xi \in \mathbb{E} \Rightarrow {f\xi } \in \mathbb{E} \] | Proof. To see this, it is enough to show that if \( f : \xi \rightarrow \tau \) is continuous and \( \mathcal{E} \in \mathfrak{E}\left( \xi \right) \) , then \( f\left( \mathcal{E}\right) \in \mathfrak{E}\left( \tau \right) \) . In fact, if \( y \in \mathop{\lim }\limits_{{f\xi }}\mathcal{F} \), then by definition ther... | Yes |
Let \( J \) be a projection preserving initial convergences (1.9) and let \( E \) be a co-projection preserving final convergences (3.6). If \( \xi \) is a JE-convergence, then the convergence quotient \( {f\xi } \) is a \( {JE} \) -convergence, hence \( J\left( {f\xi }\right) \), the \( J \) -quotient of \( \xi \) by ... | Proof. Let \( \xi \) be a \( {JE} \) -convergence on \( X \) and let \( f : X \rightarrow Y \) be surjective. We need show that \( \tau = J\left( {f\xi }\right) \) is a \( {JE} \) -convergence. Since, one always has \( {f}^{ - }{f\vartheta } \leq \vartheta \), we infer that\n\n\[ \xi \geq {JE\xi } \geq J{f}^{ - }{fE\xi... | Yes |
Theorem 5.2. Let \( E \) be equal to First, \( K \) or First \( K \) . We assume that \( \tau \) is a Hausdorff convergence and, in the case of \( E = K \) and \( E = {\operatorname{First}}_{K},\tau \) is a topologically Hausdorff pseudotopology. Then a convergence \( \tau \) is \( {JE} \) if and only if it is a \( J \... | Proof. Theorem 4.2 establishes one implication. Conversely if \( \tau \) is a \( {JE} \) -convergence, by Corollary 5.1, \( \tau \) is a \( J \) -image of \( {E\tau } \), which is an \( E \) -convergence. Moreover the separation conditions are a fortiori satisfied by \( {E\tau } \) . | Yes |
Proposition 6.2. Let \( f : X \rightarrow Y \) be surjective and let \( \xi \) be a convergence on \( X \) and \( \tau \) a Hausdorff convergence on \( Y \) . If \( f \) is compact-covering, then \( {K\tau } \geq {Sf}\left( {K\xi }\right) \) . If \( f \) is compact-first-countable-covering, then \( {\operatorname{First... | Proof. Let \( y \in \mathop{\lim }\limits_{{K\tau }}\mathcal{F} \) (respectively \( y \in \mathop{\lim }\limits_{{{\operatorname{First}}_{K}\tau }}\mathcal{F} \) ) and let \( \mathcal{U} \) be an ultrafilter of \( \mathcal{F} \) . Let \( \mathcal{K} \) be a principal (respectively countably based) \( \tau \) -compact f... | Yes |
Theorem 6.3. Let \( J \leq V \) be projections and \( E \) a co-projection. A convergence \( \tau \) is a \( {JE} \) -convergence if and only if every (continuous) \( E \) -relatively \( V \) -map onto \( \tau \) is a \( J \) -map. | Proof. Let \( \tau \geq {JE\tau } \) and (6.1); hence \( \tau \geq {JVf}\left( {E\xi }\right) = {Jf}\left( {E\xi }\right) \geq J\left( {f\xi }\right) \), because \( J \leq V \) . Vice versa, if \( {JE\tau } > \tau \), then the identity \( i : {E\tau } \rightarrow \tau \) is a continuous \( E \) -relatively \( I \) -map... | Yes |
Theorem 6.4. Let \( E \) be \( \operatorname{Seq} \), First, \( K \) or \( {\operatorname{First}}_{K} \) and let \( J \) be a projection such that \( T \leq J \leq S \) . Let \( \tau \) be a Hausdorff pseudotopology. Then the following statements are equivalent:\n\n(i) \( \tau = {JE\tau } \) ;\n\n(ii) every continuous ... | Proof. (i) \( \Rightarrow \) (ii) follows from Theorem 6.3 and (ii) \( \Rightarrow \) (iii) is obvious by Proposition 6.2. To prove (iii) \( \Rightarrow \) (i) let \( \tau \) be a Hausdorff pseudotopology on a set \( X \) and suppose that \( {JE\tau } > \tau \) . Then the identity \( i : {E\tau } \rightarrow \tau \) is... | Yes |
Theorem 1 The equation of a circle in \( {\mathbb{R}}^{2} \) with centre \( \left( {a, b}\right) \) and radius \( r \) is\n\[{\left( x - a\right) }^{2} + {\left( y - b\right) }^{2} = {r}^{2}.\] | For example, it follows from this formula that the circle with centre \( \left( {-1,2}\right) \) and radius \( \sqrt{3} \) has equation\n\[{\left( x + 1\right) }^{2} + {\left( y - 2\right) }^{2} = {\left( \sqrt{3}\right) }^{2}\]\nthis can be simplified to give\n\[{x}^{2} + {2x} + 1 + {y}^{2} - {4y} + 4 = 3\]\nor\n\[{x}... | Yes |
Theorem 2 An equation of the form\n\n\[ \n{x}^{2} + {y}^{2} + {fx} + {gy} + h = 0 \n\]\n\nrepresents a circle with\n\n\[ \n\text{centre}\left( {-\frac{1}{2}f, - \frac{1}{2}g}\right) \text{and radius}\sqrt{\frac{1}{4}{f}^{2} + \frac{1}{4}{g}^{2} - h}\text{,} \n\]\n\nprovided that \( \frac{1}{4}{f}^{2} + \frac{1}{4}{g}^{... | ## Remark\n\nIt follows from equation (4) above that if \( \frac{1}{4}{f}^{2} + \frac{1}{4}{g}^{2} - h < 0 \), then there are no points \( \left( {x, y}\right) \) that satisfy the equation \( {x}^{2} + {y}^{2} + {fx} + {gy} + h = 0 \) ; and if \( \frac{1}{4}{f}^{2} + \frac{1}{4}{g}^{2} - h = 0 \), then the given equati... | Yes |
Two intersecting circles \( {C}_{1} \) and \( {C}_{2} \) with equations\n\n\[ \n{x}^{2} + {y}^{2} + {f}_{1}x + {g}_{1}y + {h}_{1} = 0\text{ and } \n\]\n\n\[ \n{x}^{2} + {y}^{2} + {f}_{2}x + {g}_{2}y + {h}_{2} = 0, \n\]\n\nrespectively, are orthogonal if and only if\n\n\[ \n{f}_{1}{f}_{2} + {g}_{1}{g}_{2} = 2\left( {{h}... | Proof The circle \( {C}_{1} \) has centre \( A = \left( {-\frac{1}{2}{f}_{1}, - \frac{1}{2}{g}_{1}}\right) \) and radius \( {r}_{1} = \) You met these formulas in \( \sqrt{\frac{1}{4}{f}_{1}^{2} + \frac{1}{4}{g}_{1}^{2} - {h}_{1}} \) ; the circle \( {C}_{2} \) has centre \( B = \left( {-\frac{1}{2}{f}_{2}, - \frac{1}{2... | Yes |
Find the equation of the circle that passes through \( \left( {1,2}\right) \) and the points of intersection of the circles\n\n\[ \n{x}^{2} + {y}^{2} - {3x} + {4y} - 1 = 0\text{ and }{x}^{2} + {y}^{2} + \frac{5}{2}x - {3y} + \frac{3}{2} = 0.\n\] | Solution By Theorem 4, the required equation is of the form\n\n\[ \n{x}^{2} + {y}^{2} - {3x} + {4y} - 1 + k\left( {{x}^{2} + {y}^{2} + \frac{5}{2}x - {3y} + \frac{3}{2}}\right) = 0\n\]\n\n(9)\n\nfor some number \( k \) . Since \( \left( {1,2}\right) \) must satisfy this equation, it follows that\n\n\[ \n1 + 4 - 3 + 8 -... | Yes |
(a) Write down the focus, vertex, axis and directrix of \( E \) .\n\n(b) Determine the equation of the chord that joins distinct points \( P \) and \( Q \) on \( E \) with parameters \( {t}_{1} \) and \( {t}_{2} \), respectively. Determine the condition on \( {t}_{1} \) and \( {t}_{2} \) such that the chord \( {PQ} \) ... | ## Solution\n\n(a) The parabola \( E \) is the parabola in standard form where \( {4a} = 2 \), or \( a = \frac{1}{2} \).\n\n\n\nIt follows that the focus of \( E \) is \( \left( {\frac{1}{2},0}\right) \), its vertex is... | Yes |
Example 3 Let \( {PQ} \) be an arbitrary chord of the ellipse with equation\n\n\[ \frac{{x}^{2}}{{a}^{2}} + \frac{{y}^{2}}{{b}^{2}} = 1 \]\n\nLet \( M \) be the midpoint of \( {PQ} \). Prove that the following expression is independent of the choice of \( P \) and \( Q \): | Solution Let \( P \) and \( Q \) have the parametric coordinates \( \left( {a\cos {t}_{1}, b\sin {t}_{1}}\right) \) and \( \left( {a\cos {t}_{2}, b\sin {t}_{2}}\right) \), respectively. It follows that \( M \) has coordinates \( \left( {\frac{a}{2}\left( {\cos {t}_{1} + \cos {t}_{2}}\right) ,\frac{b}{2}\left( {\sin {t}... | Yes |
Let \( E \) be an ellipse with major axis \( \left( {-a, a}\right) \) and foci \( F \) and \( {F}^{\prime } \). Then, if \( P \) is a point on the ellipse, \( {FP} + P{F}^{\prime } = {2a} \). In particular, \( {FP} + P{F}^{\prime } \) is constant for all points \( P \) on the ellipse. | Proof Let \( d \) and \( {d}^{\prime } \) be the directrices of the ellipse that correspond to the foci \( F \) and \( {F}^{\prime } \), respectively. Then, since\n\n\[ \n{PF} = e \times \left( {\text{ distance from }P\text{ to }d}\right) \n\]\n\nand\n\n\[ \nP{F}^{\prime } = e \times \left( {\text{distance from}P\text{... | Yes |
Let \( H \) be a hyperbola with major axis \( \left( {-a, a}\right) \) and foci \( F \) and \( {F}^{\prime } \) . Then, if \( P \) is a point on the branch of the hyperbola that is closer to \( F \) , \[ P{F}^{\prime } - {PF} = {2a} \] and, if \( P \) is a point on the branch of the hyperbola closer to \( {F}^{\prime }... | Proof We shall prove only the first formula; the proof of the second is similar. Let \( d \) and \( {d}^{\prime } \) be the directrices of the hyperbola that correspond to the foci \( F \) and \( {F}^{\prime } \) respectively, and let \( P \) be a point on the branch of the hyperbola that is closer to \( F \) . Then, s... | Yes |
Theorem 1 The slope of the tangent to a curve in \( {\mathbb{R}}^{2} \) with parametric equations \( x = x\left( t\right), y = y\left( t\right) \) at the point with parameter \( t \) is\n\n\[ \frac{{y}^{\prime }\left( t\right) }{{x}^{\prime }\left( t\right) } \]\n\nprovided that \( {x}^{\prime }\left( t\right) \neq 0 \... | Proof The points on the curve with parameters \( t \) and \( t + h \) have coordinates \( \left( {x\left( t\right), y\left( t\right) }\right) \) and \( \left( {x\left( {t + h}\right), y\left( {t + h}\right) }\right) \), respectively. Then, if \( h \neq 0 \), the slope of the chord joining these two points is\n\n\[ \fra... | Yes |
Theorem 2 The equation of the tangent at the point \( \left( {{x}_{1},{y}_{1}}\right) \) to a conic in standard form is as follows. | Conic Tangent\n\nEllipse \( \frac{{x}^{2}}{{a}^{2}} + \frac{{y}^{2}}{{b}^{2}} = 1 \) \( \frac{x{x}_{1}}{{a}^{2}} + \frac{y{y}_{1}}{{b}^{2}} = 1 \)\n\nHyperbola \( \frac{{x}^{2}}{{a}^{2}} - \frac{{y}^{2}}{{b}^{2}} = 1 \) \( \frac{x{x}_{1}}{{a}^{2}} - \frac{y{y}_{1}}{{b}^{2}} = 1 \)\n\nParabola \( {y}^{2} = {4ax} \) \( y... | Yes |
For each of the following conics, determine the equation of the tangent to the conic at the indicated point. | We can deduce a useful fact from the equation \( x{x}_{1} + y{y}_{1} = 1 \) for the tangent at the point \( \left( {{x}_{1},{y}_{1}}\right) \) to the unit circle \( {x}^{2} + {y}^{2} = 1 \) . Let \( \left( {a, b}\right) \) be some point on this tangent, so that \[ a{x}_{1} + b{y}_{1} = 1 \] (2) Next, let the other tang... | Yes |
Problem 6 The normal to the parabola with parametric equations\n\n\n\n\\( x = {t}^{2}, y = {2t}\\left( {t \\in \\mathbb{R}}\\right) \\) at the point \\( P \\) with parameter \\( t, t \\neq 0 \\), meets the parabola at ... | (a) Prove that the slope of the normal to the parabola at \\( P \\) is \\( - t \\) .\n\n(b) Find the equation of the normal to the parabola at \\( P \\) .\n\n(c) By substituting the coordinates of \\( Q \\) into your equation from part (b), prove that \\( T = - \\frac{2}{t} - t \\) . | Yes |
Theorem 4 A perpendicular from a focus of a non-degenerate conic to a tangent meets the tangent on the auxiliary circle of the conic. | Proof (for a parabola) Let the point \( P\left( {a{t}^{2},{2at}}\right) \) lie on the parabola in standard form with equation \( {y}^{2} = {4ax} \), and let the perpendicular from the focus \( F\left( {a,0}\right) \) to the tangent at \( P \) meet it at \( T \) .\n\nBy Theorem 2 of Subsection 1.2.1, the tangent at \( P... | Yes |
Theorem 1 Any conic has an equation of the form\n\n\[ A{x}^{2} + {Bxy} + C{y}^{2} + {Fx} + {Gy} + H = 0, \]\n\nwhere \( A, B, C, F, G \) and \( H \) are real numbers, and not all of \( A, B \) and \( C \) are zero. Conversely, any set of points in \( {\mathbb{R}}^{2} \) whose coordinates \( \left( {x, y}\right) \) sati... | We omit a proof of the converse part. It would simply be a reworking of the classification methods in the rest of the section. This will be useful, since we can then use the whole armoury of Linear Algebra to study such equations. Here we choose to regard \( 1 \times 1 \) matrices and real numbers as equivalent; this w... | No |
Theorem 2 A \( 2 \times 2 \) matrix \( \mathbf{P} \) represents a rotation of \( {\mathbb{R}}^{2} \) about the origin If \( \mathbf{P} \) is orthogonal, then if and only if it satisfies the following two conditions: det \( \mathbf{P} = \pm 1 \) ; when \( \det \mathbf{P} = - 1,\mathbf{P} \) represents\n\n(a) \( \mathbf{... | Proof A matrix \( \mathbf{P} \) represents a rotation about the origin (anticlockwise through an angle \( \theta \) ) if and only it is of the form\n\n\[ \left( \begin{array}{ll} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array}\right) \]\n\n(6)\n\nIt is easy to verify that \( \mathbf{P} \) satisfies... | Yes |
Theorem 3 A non-degenerate conic with equation \[ A{x}^{2} + {Bxy} + C{y}^{2} + {Fx} + {Gy} + H = 0 \] and matrix \( \mathbf{A} = \left( \begin{matrix} A & \frac{1}{2}B \\ \frac{1}{2}B & C \end{matrix}\right) \) can be classified as follows: Since \( \det \mathbf{A} = {AC} - \) \( \frac{1}{4}{B}^{2} = - \frac{1}{4}\lef... | We omit a proof of this result. | No |
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