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Theorem 7.9 (Morera’s Theorem). Let \( U \) be an open set in \( \mathbf{C} \) and let \( f \) be continuous on \( U \) . Assume that the integral of \( f \) along the boundary of every closed rectangle contained in \( U \) is 0 . Then \( f \) is analytic. | Proof. By Theorem 3.2, we know that \( f \) has a local primitive \( g \) at every point on \( U \), and hence that \( g \) is holomorphic. By Theorem 7.2, we conclude that \( g \) is analytic, and hence that \( {g}^{\prime } = f \) is analytic, as was to be shown. | Yes |
Lemma 1.1. If \( \gamma \) is a closed path, then \( W\left( {\gamma ,\alpha }\right) \) is an integer. | Proof. Let \( \gamma = \left\{ {{\gamma }_{1},\ldots ,{\gamma }_{n}}\right\} \) where each \( {\gamma }_{i} \) is a curve defined on an interval \( \left\lbrack {{a}_{i},{b}_{i}}\right\rbrack \) . After a reparametrization of each curve if necessary, we may assume without loss of generality that \( {b}_{i} = {a}_{i + 1... | Yes |
Lemma 1.2. Let \( \gamma \) be a path. Then the function of \( \alpha \) defined by\n\n\[ \alpha \mapsto {\int }_{\gamma }\frac{1}{z - \alpha }{dz} \]\n\nfor \( \alpha \) not on the path, is a continuous function of \( \alpha \) . | Proof. Given \( {\alpha }_{0} \) not on the path, we have to see that\n\n\[ {\int }_{\gamma }\left( {\frac{1}{z - \alpha } - \frac{1}{z - {\alpha }_{0}}}\right) {dz} \]\n\ntends to 0 as \( \alpha \) tends to \( {\alpha }_{0} \) . This integral is estimated as follows. The function \( t \mapsto \left| {{\alpha }_{0} - \... | Yes |
Lemma 1.3. Let \( \gamma \) be a closed path. Let \( S \) be a connected set not intersecting \( \gamma \) . Then the function\n\n\[ \n\alpha \mapsto \frac{1}{2\pi i}{\int }_{\gamma }\frac{1}{z - \alpha }{dz}\n\]\n\nis constant for \( \alpha \) in \( S \) . If \( S \) is not bounded, then this constant is 0 . | Proof. We know from Lemma 1.1 that the integral is the winding number, and is therefore an integer. If a function takes its values in the integers, and is continuous, then it is constant on any curve, and consequently constant on a connected set. If \( S \) is not bounded, then for \( \alpha \) arbitrarily large, the i... | Yes |
(i) If \( \gamma ,\eta \) are closed paths in \( U \) and are homotopic, then they are homologous. | Proof. The first statement follows from Theorem 5.2 of the preceding chapter because the function \( 1/\left( {z - \alpha }\right) \) is analytic on \( U \) for \( \alpha \notin U \) . | No |
Corollary 2.3. If \( \gamma ,\eta \) are closed chains in \( U \) and \( \gamma ,\eta \) are homologous in \( U \), then\n\n\[{\int }_{\gamma }f = {\int }_{\eta }f\] | Proof. Apply Cauchy’s theorem to the closed chain \( \gamma - \eta \) . | No |
Theorem 2.5 (Cauchy’s Formula). Let \( \gamma \) be a closed chain in \( U \), homologous to 0 in \( U \). Let \( f \) be analytic on \( U \), let \( {z}_{0} \) be in \( U \) and not on \( \gamma \). Then\n\n\[ \frac{1}{2\pi i}{\int }_{\gamma }\frac{f\left( z\right) }{z - {z}_{0}}{dz} = W\left( {\gamma ,{z}_{0}}\right)... | Proof. We base this proof on Theorems 2.2 and 2.4. An independent proof will be given below. By assumption, in a neighborhood of \( {z}_{0} \), we have a power series expansion\n\n\[ f\left( z\right) = {a}_{0} + {a}_{1}\left( {z - {z}_{0}}\right) + \text{ higher terms,}\;\text{ with }{a}_{0} = f\left( {z}_{0}\right) .\... | Yes |
Lemma 3.1. Let \( \gamma \) be a path in an open set \( U \) . Then there exists a rectangular path \( \eta \) with the same end points, and such that \( \gamma ,\eta \) are close together in \( U \) in the sense of Chapter III,§4. In particular, \( \gamma \) and \( \eta \) are homologous in \( U \), and for any holomo... | Proof. Suppose \( \gamma \) is defined on an interval \( \left\lbrack {a, b}\right\rbrack \) . We take a partition of the interval, \[ a = {a}_{0} \leqq {a}_{1} \leqq {a}_{2} \leqq \cdots \leqq {a}_{n} = b \] such that the image of each small interval \[ \gamma \left( \left\lbrack {{a}_{i},{a}_{i + 1}}\right\rbrack \ri... | Yes |
Theorem 1.1. Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of holomorphic functions on an open set \( U \) . Assume that for each compact subset \( K \) of \( U \) the sequence converges uniformly on \( K \), and let the limit function be \( f \) . Then \( f \) is holomorphic. | Proof. Let \( {z}_{0} \\in U \), and let \( {\\bar{D}}_{R} \) be a closed disc of radius \( R \) centered at \( {z}_{0} \) and contained in \( U \) . Then the sequence \( \\left\\{ {f}_{n}\\right\\} \) converges uniformly on \( {\\bar{D}}_{R} \) . Let \( {C}_{R} \) be the circle which is the boundary of \( {\\bar{D}}_{... | Yes |
Theorem 1.2. Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of analytic functions on an open set \( U \), converging uniformly on every compact subset \( K \) of \( U \) to a function \( f \) . Then the sequence of derivatives \( \\left\\{ {f}_{n}^{\\prime }\\right\\} \) converges uniformly on every compact subset... | Proof. The proof will be left as an exercise to the reader. [Hint: Cover the compact set with a finite number of closed discs contained in \( U \), and of sufficiently small radius. Cauchy’s formula expresses the derivative \( {f}_{n}^{\\prime } \) as an integral, and one can argue as in the previous theorem.] | No |
Theorem 2.1. Let \( A \) be the above annulus, and let \( f \) be a holomorphic function on \( A \) . Let \( r < s < S < R \) . Then \( f \) has a Laurent expansion\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{a}_{n}{z}^{n} \]\n\nwhich converges absolutely and uniformly on \( s \leqq \left... | Proof. For some \( \epsilon > 0 \) we may assume (by the definition of what it means for \( f \) to be holomorphic on the closed annulus) that \( f \) is holomorphic on the open annulus \( U \) of complex numbers \( z \) such that\n\n\[ r - \epsilon < \left| z\right| < R + \epsilon \]\n\nThe chain \( {C}_{R} - {C}_{r} ... | Yes |
Theorem 3.1. If \( f \) is bounded in some neighborhood of \( {z}_{0} \), then one can define \( f\left( {z}_{0}\right) \) in a unique way such that the function is also analytic at \( {z}_{0} \) . | Proof. Say \( {z}_{0} = 0 \) . By \( §2 \), we know that \( f \) has a Laurent expansion\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n \geqq 0}}{a}_{n}{z}^{n} + \mathop{\sum }\limits_{{n < 0}}{a}_{n}{z}^{n} \]\n\nfor \( 0 < \left| z\right| < R \) . We have to show \( {a}_{n} = 0 \) if \( n < 0 \) . Let \( n = - m ... | Yes |
Theorem 3.2 (Casorati-Weierstrass). Let 0 be an essential singularity of the function \( f \), and let \( D \) be a disc centered at 0 on which \( f \) is holomorphic except at 0 . Let \( U \) be the complement of 0 in \( D \) . Then \( f\left( U\right) \) is dense in the complex numbers. In other words, the values of ... | Proof. Suppose the theorem is false. There exists a complex number \( \alpha \) and a positive number \( s > 0 \) such that\n\n\[ \left| {f\left( z\right) - \alpha }\right| > s\;\text{ for all }z \in U. \]\n\nThe function\n\n\[ g\left( z\right) = \frac{1}{f\left( z\right) - \alpha } \]\n\nis then holomorphic on \( U \)... | Yes |
Theorem 3.3. The only analytic automorphisms of \( \mathbf{C} \) are the functions of the form \( f\left( z\right) = {az} + b \), where \( a, b \) are constants, \( a \neq 0 \) . | Proof. Let \( f \) be an analytic automorphism of \( \mathbf{C} \) . After making a translation by \( - f\left( 0\right) \), we may assume without loss of generality that \( f\left( 0\right) = 0 \) . We then have to prove that \( f\left( z\right) = {az} \) for some constant \( a \) . Let\n\n\[ h\left( z\right) = f\left... | Yes |
Theorem 1.1. Let \( {z}_{0} \) be an isolated singularity of \( f \), and let \( C \) be a small circle oriented counterclockwise, centered at \( {z}_{0} \) such that \( f \) is holomorphic on \( C \) and its interior, except possibly at \( {z}_{0} \) . Then\n\n\[{\int }_{C}f\left( \zeta \right) {d\zeta } = {2\pi i}{a}... | Proof. Since the series for \( f\left( \zeta \right) \) converges uniformly and absolutely for \( \zeta \) on the circle, we may integrate it term by term. The integral of \( {\left( \zeta - {z}_{0}\right) }^{n} \) over the circle is equal to 0 for all values of \( n \) except possibly when \( n = - 1 \), in which case... | Yes |
Theorem 1.2 (Residue Formula). Let \( U \) be an open set, and \( \gamma \) a closed chain in \( U \) such that \( \gamma \) is homologous to 0 in \( U \) . Let \( f \) be analytic on \( U \) except at a finite number of points \( {z}_{1},\ldots ,{z}_{n} \) . Let \( {m}_{i} = W\left( {\gamma ,{z}_{i}}\right) \) . Then\... | Proof. Immediate by plugging Theorem 1.1 in the above mentioned theorem of Chapter IV. | No |
Theorem 1.5. Let \( \gamma \) be a closed chain in \( U \), homologous to 0 in \( U \). Let \( f \) be meromorphic on \( U \), with only a finite number of zeros and poles, say at the points \( {z}_{1},\ldots ,{z}_{n} \), none of which lie on \( \gamma \). Let \( {m}_{i} = W\left( {\gamma ,{z}_{i}}\right) \). Then\n\n\... | Proof. This is immediate by plugging the statement of the lemma into the residue formula. | No |
Theorem 1.6 (Rouché’s Theorem). Let \( \gamma \) be a closed path homologous to 0 in \( U \) and assume that \( \gamma \) has an interior. Let \( f, g \) be analytic on \( U \) , and \[ \left| {f\left( z\right) - g\left( z\right) }\right| < \left| {f\left( z\right) }\right| \] for \( z \) on \( \gamma \) . Then \( f \)... | Proof. Note that the assumption implies automatically that \( f, g \) have no zero on \( \gamma \) . We have \[ \left| {\frac{g\left( z\right) }{f\left( z\right) } - 1}\right| < 1 \] for \( z \) on \( \gamma \) . Then the values of the function \( g/f \) are contained in the open disc with center 1 and radius 1 . Let \... | Yes |
Theorem 1.8. Let \( w \) be a local coordinate at \( {z}_{0} \) . Let \( \omega \) be a meromorphic differential in a neighborhood of \( {z}_{0} \), and write \( \omega = f\left( z\right) {dz} = \) \( g\left( w\right) {dw} \), where \( f, g \) are meromorphic functions, with the power series expansions as in (1) and (4... | Proof. Let \( \gamma \) be a small circle around \( {z}_{0} \) in the \( z \) -plane. Let \( w = \varphi \left( z\right) \) .\n\nThen\n\n\[ \n{b}_{-1} = \frac{1}{2\pi i}{\int }_{{\varphi }_{ \circ }\gamma }g\left( w\right) {dw} = \frac{1}{2\pi i}{\int }_{\gamma }f\left( z\right) {dz} = {a}_{-1}, \n\]\n\nwhich proves th... | Yes |
Theorem 2.1. Suppose that there exists a number \( B > 0 \) such that for all \( \left| z\right| \) sufficiently large, we have\n\n\[ \left| {f\left( z\right) }\right| \leqq B/{\left| z\right| }^{2} \]\n\nThen\n\n\[ \mathop{\lim }\limits_{{R \rightarrow \infty }}{\int }_{{S}_{R}}f = 0 \]\n\nand the above formula is val... | Proof. The integral is estimated by the sup norm of \( f \), which is \( B/{R}^{2} \) by assumption, multiplied by the length of the semicircle, which is \( {\pi R} \) . Since \( {\pi B}/R \) tends to 0 as \( R \rightarrow \infty \), our theorem is proved. | Yes |
Theorem 2.2. Let \( f \) be meromorphic on \( \mathbf{C} \), having only a finite number of poles, not lying on the real axis. Suppose that there is a constant \( K \) such that\n\n\[ \left| {f\left( z\right) }\right| \leqq K/\left| z\right| \]\n\nfor all sufficiently large \( \left| z\right| \) . Let \( a > 0 \) . The... | Proof. For simplicity, take \( a = 1 \) . We integrate over any rectangle as shown on Fig. 8, taking \( T = A + B \) . Taking \( A, B > 0 \) sufficiently large, it suffices to prove that the integral over the three sides other than the bottom side tend to 0 as \( A, B \) tend to infinity.\n\n![61158264-2ef3-4585-a05d-6... | Yes |
Theorem 2.3. Let \( Q\left( {x, y}\right) \) be a rational function which is continuous when \( {x}^{2} + {y}^{2} = 1 \) . Let \( f\left( z\right) \) be as above. Then\n\n\( {\int }_{0}^{2\pi }Q\left( {\cos \theta ,\sin \theta }\right) {d\theta } = {2\pi i}\sum \) residues of \( f \) inside the unit circle. | Proof. Let \( C \) be the unit circle. Then\n\n\[ \n{\int }_{C}f\left( z\right) {dz} = {2\pi i}\sum \text{residues of}f\text{inside the circle.} \n\]\n\nOn the other hand, by definition the integral on the left is equal to\n\n\[ \n{\int }_{0}^{2\pi }f\left( {e}^{i\theta }\right) i{e}^{i\theta }{d\theta } = {\int }_{0}^... | Yes |
Theorem 1.1. Let \( f : D \rightarrow D \) be an analytic function of the unit disc into itself such that \( f\left( 0\right) = 0 \) . Then:\n\n(i) We have \( \left| {f\left( z\right) }\right| \leqq \left| z\right| \) for all \( z \in D \) .\n\n(ii) If for some \( {z}_{0} \neq 0 \) we have \( \left| {f\left( {z}_{0}\ri... | Proof. Let\n\n\[ f\left( z\right) = {a}_{1}z + \cdots \]\n\nbe the power series for \( f \) . The constant term is 0 because we assumed \( f\left( 0\right) = 0 \) . Then \( f\left( z\right) /z \) is holomorphic, and\n\n\[ \left| \frac{f\left( z\right) }{z}\right| < 1/r\;\text{ for }\;\left| z\right| = r < 1 \]\n\nconse... | Yes |
Theorem 1.2. Let \( f : D \rightarrow D \) be an analytic function of the unit disc into itself such that \( f\left( 0\right) = 0 \) . Let\n\n\[ f\left( z\right) = {a}_{1}z + \text{higher terms.} \]\n\nThen \( \left| {{f}^{\prime }\left( 0\right) }\right| = \left| {a}_{1}\right| \leqq 1 \), and if \( \left| {a}_{1}\rig... | Proof. Since \( f\left( 0\right) = 0 \), the function \( f\left( z\right) /z \) is analytic at \( z = 0 \), and\n\n\[ \frac{f\left( z\right) }{z} = {a}_{1} + \text{ higher terms. } \]\n\nLetting \( z \) approach 0 and using the first part of Theorem 1.1 shows that \( \left| {a}_{1}\right| \leqq 1 \) . Next suppose \( \... | Yes |
Theorem 2.1. Let \( f : D \rightarrow D \) be an analytic automorphism of the unit disc and suppose \( f\left( \alpha \right) = 0 \) . Then there exists a real number \( \varphi \) such that\n\n\[ f\left( z\right) = {e}^{i\varphi }\frac{\alpha - z}{1 - \bar{\alpha }z}. \] | Proof. Let \( g = {g}_{\alpha } \) be the above automorphism. Then \( f \circ {g}^{-1} \) is an automorphism of the unit disc, and maps 0 on 0, i.e. it has a zero at 0 . It now suffices to prove that the function \( h\left( w\right) = f\left( {{g}^{-1}\left( w\right) }\right) \) is of the form\n\n\[ h\left( w\right) = ... | Yes |
Corollary 2.2. If \( f \) is an automorphism of the disc which leaves the origin fixed, i.e. \( f\left( 0\right) = 0 \), then \( f\left( z\right) = {e}^{i\varphi }z \) for some real number \( \varphi \), so \( f \) is a rotation. | Proof. Let \( \alpha = 0 \) in the theorem. | Yes |
Theorem 3.1. Let \( H \) be the upper half plane. The map\n\n\[ f : z \mapsto \frac{z - i}{z + i} \]\n\n is an isomorphism of \( H \) with the unit disc. | Proof. Let \( w = f\left( z\right) \) and \( z = x + {iy} \) . Then\n\n\[ f\left( z\right) = \frac{x + \left( {y - 1}\right) i}{x + \left( {y + 1}\right) i}. \]\n\nSince \( z \) is in \( H, y > 0 \), it follows that \( {\left( y - 1\right) }^{2} < {\left( y + 1\right) }^{2} \) whence\n\n\[ {x}^{2} + {\left( y - 1}\righ... | Yes |
Isomorphism between quarter disc and half disc | Figure 6 | No |
Example 3. | \n\nUpper half disc with first quadrant\n\nFigure 7 | No |
We want to get an isomorphism of \( U \) with the upper half plane. | All we have to do is to compose the isomorphisms of Examples 2, 3, and 1 in that order. Thus an isomorphism of \( U \) with \( H \) is given by the formula in the picture. | Yes |
Example 10. | \n\nFigure 14 | No |
In this example, the obstacle is a bump rather than a vertical line segment. We claim that the map \[ z \mapsto z + \frac{1}{z} \] is an isomorphism of the open set \( U \) lying inside the upper half plane, above the unit circle, with the upper half plane. | Proof. Let \( w = z + 1/z \) so that \[ w = x\left( {1 + \frac{1}{{x}^{2} + {y}^{2}}}\right) + {iy}\left( {1 - \frac{1}{{x}^{2} + {y}^{2}}}\right) . \] If \( z \in U \), then \( \left| z\right| > 1 \) so \( \operatorname{Im}w > 0 \) and \( w \in H \) . The quadratic equation \[ {z}^{2} - {zw} + 1 = 0 \] has two distinc... | Yes |
The sine function maps the interval \( \left\lbrack {-\pi /2,\pi /2}\right\rbrack \) on the interval \( \left\lbrack {-1,1}\right\rbrack \) . | Let us look also at what the sine does to the right vertical boundary, which consists of all points \( \pi /2 + {it} \) with \( t \geqq 0 \) . We know that\n\n\[ \sin z = \frac{{e}^{iz} - {e}^{-{iz}}}{2i} \]\n\nHence\n\n\[ \sin \left( {\frac{\pi }{2} + {it}}\right) = \frac{{e}^{{i\pi }/2}{e}^{iit} - {e}^{-{i\pi }/2}{e}... | Yes |
Theorem 4.1. Let \( U \) be a bounded connected open set, \( \bar{U} \) its closure. Let \( f \) be a continuous function on \( \bar{U} \), analytic on \( U \) . Suppose that \( f \) is not constant, and maps the boundary of \( U \) into the unit circle, so\n\n\[ \left| {f\left( z\right) }\right| = 1\;\text{ for all }z... | Proof. That \( f \) maps \( U \) into \( D \) follows from the maximum modulus principle. Suppose there exists some \( \alpha \in D \) but \( \alpha \) is not in the image of \( f \) . Let \( {g}_{\alpha } \) be the automorphism of \( D \) interchanging 0 and \( \alpha \), given in Chapter VII, \( §2 \), so \( {g}_{\al... | Yes |
Lemma 4.2. Let \( \gamma \) be a piecewise \( {C}^{1} \) closed path in an open set \( U \) of \( \mathbf{C} \). Suppose that \( \gamma \) has an interior, denoted by \( \operatorname{Int}\left( \gamma \right) \). Then the union \[ S = \operatorname{Int}\left( \gamma \right) \cup \gamma \] (identifying \( \gamma \) wit... | Proof. To see this, we have to prove that \( S \) is closed and bounded. Let \( \left\langle {z}_{n}\right\rangle \) be a sequence in \( S \), converging to some point in \( \mathbf{C} \). If the sequence contains infinitely many points of \( \operatorname{Int}\left( \gamma \right) \), then for such points \( W\left( {... | Yes |
Theorem 4.3. Let \( \gamma \) be a piecewise \( {C}^{1} \) closed path in a connected open set \( U \) of \( \mathbf{C} \) . Assume \( \gamma \) homologous to 0 in \( U \) . Let \( f \) be analytic nonconstant on \( U \) . Assume that \( \gamma \) and \( f \circ \gamma \) have interiors and \( f \circ \gamma \) does no... | Proof. For \( \alpha \in \operatorname{Int}\left( \gamma \right) \), let \( {f}_{\alpha }\left( z\right) = f\left( z\right) - f\left( \alpha \right) \) . Then by the chain rule:\n\n\[ \left. {W\left( {f \circ \gamma }\right), f\left( \alpha \right) }\right) = \frac{1}{2\pi i}{\int }_{f \circ \gamma }\frac{1}{\zeta - f\... | Yes |
Theorem 5.1. Given a fractional linear map \( F \), there exist complex numbers \( \alpha ,\beta ,\gamma \) such that either \( F = {\alpha z} + \beta \), or\n\n\[ F\left( z\right) = {T}_{\gamma } \circ {M}_{\alpha } \circ J \circ {T}_{\beta } \] | Proof. Suppose \( c = 0 \) . Then \( F\left( z\right) = \left( {{az} + b}\right) /d \) and \( F = {T}_{\beta } \circ {M}_{\alpha } \), with \( \beta = b/d,\alpha = a/d \) . Suppose this is not the case, so \( c \neq 0 \) . We divide \( a, b \) , \( c, d \) by \( c \) and using these new numbers gives the same map \( F ... | Yes |
Theorem 5.2. A fractional linear transformation maps straight lines and circles onto straight lines and circles. (Of course, a circle may be mapped onto a line and vice versa.) | Proof. By Theorem 5.1 it suffices to prove the assertion in each of the three cases of the simple maps. The assertion is obvious for translations and multiplications (which are rotations followed by dilations). There remains to deal with the inversion.\n\n be fractional linear map. If \( \infty \) is a fixed point of \( F \), then there exist complex numbers \( a, b \) such that \( F\left( z\right) = {az} + b \) . | Proof. Let \( F\left( z\right) = \left( {{az} + b}\right) /\left( {{cz} + d}\right) \) . If \( c \neq 0 \) then \( F\left( \infty \right) = a/c \) which is not \( \infty \) . By hypothesis, it follows that \( c = 0 \), in which case the assertion is obvious. | Yes |
Theorem 5.4. Given any three distinct points \( {z}_{1},{z}_{2},{z}_{3} \) on the Riemann sphere, and any three distinct points \( {w}_{1},{w}_{2},{w}_{3} \), there exists a unique fractional linear map \( F \) such that\n\n\[ F\left( {z}_{i}\right) = {w}_{i}\;\text{ for }\;i = 1,2,3. \] | Proof. We proceed stepwise, and first prove uniqueness. Let \( F, G \) be fractional linear maps which have the same effect on three points. Then \( F \circ {G}^{-1} \) has three fixed points, and it suffices to prove the following lemma.\n\nLemma 5.5. Let \( F \) | No |
Lemma 5.5. Let \( F \) be a fractional linear map. If \( F \) has three fixed points, then \( F \) is the identity. | Proof. Suppose first that one fixed point is \( \infty \) . By Proposition 5.3, we know that \( F\left( z\right) = {az} + b \) . Suppose \( {z}_{1} \in \mathbf{C} \) and \( {z}_{1} \) is a fixed point. Then \( a{z}_{1} + b = {z}_{1} \) so \( \left( {1 - a}\right) {z}_{1} = b \) . If \( a \neq 1 \) then we see that \( {... | No |
Theorem 5.6. The function\n\n\\[ \n z \\mapsto \\frac{z - {z}_{1}}{z - {z}_{2}}\\frac{{z}_{3} - {z}_{2}}{{z}_{3} - {z}_{1}} \n\\]\n\n is the unique function such that \\( F\\left( {z}_{1}\\right) = 0, F\\left( {z}_{2}\\right) = \\infty, F\\left( {z}_{3}\\right) = 1 \\) . If \\( w = F\\left( z\\right) \\) is the functio... | Example. Find the map \\( F \\) in Theorem 5.4 such that\n\n\\[ \nF\\left( 1\\right) = i,\\;F\\left( i\\right) = - 1,\\;F\\left( {-1}\\right) = 1. \n\\]\n\nBy the formula,\n\n\\[ \n\\frac{w - i}{w + 1}\\frac{1 + 1}{1 - i} = \\frac{z - 1}{z - i}\\frac{-1 - i}{-1 - 1} \n\\]\n\nor in other words,\n\n\\[ \n\\frac{w - i}{w ... | Yes |
Theorem 1.2. Let \( g \) be a primitive for \( f \) on \( U \), that is, \( {g}^{\prime } = f \) . Write \( g \) in terms of its real and imaginary parts,\n\n\[ g = \varphi + {i\psi } \]\n\nThen \( \varphi \) is a potential function for \( \bar{F} \) . | Proof. Go back to Chapter I,§6. By definition, \( {g}^{\prime } = u + {iv} \) . The first computation of that section shows that\n\n\[ \frac{\partial \varphi }{\partial x} = u\;\text{ and }\;\frac{\partial \varphi }{\partial y} = - v \]\n\nas desired. | No |
Theorem 1.3. Let \( U \) be a bounded open set. Let \( u, v \) be two continuous functions on the closure \( \bar{U} \) of \( U \), and assume that \( u, v \) are harmonic on U. Assume that \( u = v \) on the boundary of \( U \) . Then \( u = v \) on \( U \) . | Proof. Subtracting the two harmonic functions having the same boundary value yields a harmonic function with boundary value 0 . Let \( u \) be such a function. We have to prove that \( u = 0 \) . Suppose there is a point \( \left( {{x}_{0},{y}_{0}}\right) \in U \) such that \( u\left( {{x}_{0},{y}_{0}}\right) > 0 \) . ... | Yes |
Theorem 3.1. Let \( U \) be a simply connected open set. Let \( u \) be harmonic on \( U \) . Then there exists an analytic function \( f \) on \( U \) such that \( u = \) \( \operatorname{Re}f \) . The difference of two such functions is a pure imaginary constant. | Proof. Let\n\n\[ h = 2\frac{\partial u}{\partial z} = \frac{\partial u}{\partial x} - i\frac{\partial u}{\partial y} \]\n\nThen \( h \) has continuous partials of first order. Furthermore \( h \) is analytic, because\n\n\[ \frac{\partial h}{\partial \bar{z}} = 2\frac{\partial }{\partial \bar{z}}\frac{\partial }{\partia... | Yes |
Lemma 3.2. Let \( f, g \) be analytic functions on a connected open set \( U \) . Suppose \( f, g \) have the same real part. Then \( f = g + {iK} \) for some real constant \( K \) . | Proof. Considering \( f - g \), it suffices to prove that if the real part of an analytic function \( f \) on \( U \) is 0 then the function is a pure imaginary constant. But this is immediate from the open mapping Theorem 6.2 of Chapter II, because \( f \) cannot map an open set on a straight line, hence \( f \) is co... | Yes |
Theorem 3.3 (Mean Value Theorem). Let \( u \) be a harmonic function on an open set \( U \) . Let \( {z}_{0} \in U \), and let \( r > 0 \) be a number such that the closed disc of radius \( r \) centered at \( {z}_{0} \) is contained in \( U \) . Then\n\n\[ u\left( {z}_{0}\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }u\l... | Proof. There is a number \( {r}_{1} > r \) such that the disc of radius \( {r}_{1} \) centered at \( {z}_{0} \) is contained in \( U \) . Any \( {r}_{1} > r \) and close to \( r \) will do. By Theorem 3.1, there is an analytic function \( f \) on the disc of radius \( {r}_{1} \) such that \( u = \operatorname{Re}f \) .... | Yes |
Theorem 3.5 (Hadamard Three-Circle Theorem). Let \( f \) be holomorphic on a closed annulus \( 0 < {r}_{1} < \left| z\right| < {r}_{2} \) . Let\n\n\[ s = \frac{\log {r}_{1} - \log r}{\log {r}_{2} - \log {r}_{1}} \]\n\nLet \( M\left( r\right) = {M}_{f}\left( r\right) = \parallel f{\parallel }_{r} = \max \left| {f\left( ... | Proof. Let \( \alpha \) be a real number. The function \( \alpha \log \left| z\right| + \log \left| {f\left( z\right) }\right| \) is harmonic outside the zeros of \( f \) . Near the zeros of \( f \) the above function has values which are large negative. Hence by the maximum modulus principle this function has its maxi... | Yes |
Theorem 3.6. Let \( u \) be harmonic in an annulus \( 0 < {r}_{1} < r < {r}_{2} \) . Then there exist constants \( a, b \) such that\n\n\[{\int }_{0}^{2\pi }u\left( {r,\theta }\right) \frac{d\theta }{2\pi } = a\log r + b\] | Proof. We shall use elementary properties of Fourier series with which readers are likely to be acquainted. For each integer \( n \) let\n\n\[{u}_{n}\left( r\right) = {\int }_{0}^{2\pi }u\left( {r,\theta }\right) {e}^{-{in\theta }}\frac{d\theta }{2\pi }\]\n\nThus \( {u}_{n}\left( r\right) \) is the \( n \) -th Fourier ... | Yes |
Theorem 3.7. Let \( U \) be an annulus \( 0 \leqq {r}_{1} < \left| z\right| < {r}_{2} \) (with \( {r}_{2} \) possibly equal to \( \infty \) ). Let \( u \) be harmonic on \( U \) . Then there exists a real constant a and an analytic function \( g \) on \( U \) such that\n\n\[ u - a\log r = \operatorname{Re}\left( g\righ... | Proof. We consider the half annuli as illustrated on Figure 15.\n\n\n\nFigure 15\n\nBy Theorem 3.1, there exists an analytic function \( {f}_{1} \) on the upper half annulus such that \( \operatorname{Re}\left( {f}_{... | Yes |
Theorem 3.8. Let \( u \) be a harmonic function on the punctured disc \( {D}^{ * } \) . Assume that \( u \) is bounded. Then \( u \) extends to a harmonic function on \( D \) . | Proof. By Theorem 3.7 there exists an analytic function \( g \) on \( {D}^{ * } \) such that \( u\left( z\right) = \operatorname{Re}\left( {g\left( z\right) }\right) + a\log r \) . We then have the Laurent expansion\n\n\[ g\left( z\right) = \mathop{\sum }\limits_{{-\infty }}^{\infty }{a}_{n}{z}^{n} \]\n\nand it suffice... | Yes |
Lemma 3.10. Let \( U \) be a connected open set. Let \( u \) be harmonic on \( U \) , and \( f \) analytic on \( U \) . If \( u = \operatorname{Re}\left( f\right) \) on some open disc contained in \( U \) , then \( u = \operatorname{Re}\left( f\right) \) on \( U \) . | Proof. Let \( V \) be the union of all open subsets of \( U \) where \( u = \operatorname{Re}\left( f\right) \) . Then \( V \) is not empty and is open. We need only show that \( V \) is closed in \( U \) . Let \( \left\{ {z}_{n}\right\} \) be a sequence of points in \( V \) converging to a point \( w \in U \) . Let \(... | Yes |
Theorem 4.1. Let \( f \) be holomorphic on the closed disc \( {\bar{D}}_{R} \) . Let \( z \in {D}_{R} \) . Then\n\n\[ f\left( z\right) = {\int }_{0}^{2\pi }f\left( {R{e}^{i\theta }}\right) \operatorname{Re}\frac{R{e}^{i\theta } + z}{R{e}^{i\theta } - z}\frac{d\theta }{2\pi }.\] | Proof. Write \( z = r{e}^{i\varphi } \) . Let \( {C}_{R} \) denote the circle of radius \( R \) , parametrized by \( \zeta = R{e}^{i\theta },{d\zeta } = {iR}{e}^{i\theta }{d\theta } \) . Then by Cauchy’s theorem,\n\n\[ f\left( z\right) = \frac{1}{2\pi i}{\int }_{{C}_{R}}\frac{f\left( \zeta \right) }{\zeta - z}{d\zeta }... | Yes |
Theorem 4.2. Let \( f \) be holomorphic on the closed disc \( {\bar{D}}_{R} \), then there is a real constant \( K \) such that for all \( z \in {D}_{R} \) we have\n\n\[ f\left( z\right) = {\int }_{0}^{2\pi }\operatorname{Re}f\left( {R{e}^{i\theta }}\right) \frac{R{e}^{i\theta } + z}{R{e}^{i\theta } - z}\frac{d\theta }... | Proof. The right-hand side is analytic in \( z \) . One can see this either by applying Theorem 7.7 of Chapter III, or by differentiating under the integral sign using Theorem A3 of \( \$ 6 \), which justifies such differentiation. By Theorem 4.1, the right-hand side and the left-hand side, namely \( f \) , have the sa... | Yes |
Theorem 5.1. Let \( f \) be continuous periodic. Then the sequence \( \left\{ {{K}_{n} * f}\right\} \) converges to \( f \) uniformly. | Proof. Changing variables, we have\n\n\[ \n{f}_{n}\left( x\right) = {\int }_{-\pi }^{\pi }f\left( {x - t}\right) {K}_{n}\left( t\right) {dt} \n\]\n\nOn the other hand, by DIR 2,\n\n\[ \nf\left( x\right) = f\left( x\right) {\int }_{-\pi }^{\pi }{K}_{n}\left( t\right) {dt} = {\int }_{-\pi }^{\pi }f\left( x\right) {K}_{n}... | Yes |
Theorem 5.2. Let \( f \) be a real valued continuous function, periodic of period \( {2\pi } \) . Then there exists a function \( u \), continuous on the closed disc and harmonic on the open disc, such that \( u = f \) on the circle, in other words \( u\left( {1,\theta }\right) = f\left( \theta \right) \) . This functi... | Proof. The Laplace operator in polar coordinates can be applied to \( {P}_{r}\left( \theta \right) \), differentiating the series term by term, which is obviously allowable. If you do this, you will find that\n\n\[ \left\lbrack {{r}^{2}\left( \frac{{\partial }^{2}}{\partial {r}^{2}}\right) + r\left( \frac{\partial }{\p... | Yes |
Theorem 5.4. Let \( u \) be continuous on the closed unit disc \( \bar{D} \), and harmonic on the disc D. Then there exists an analytic function \( f \) on D such that \( u = \operatorname{Re}f \), and two such functions differ by a pure imaginary constant. In fact,\n\n\[ f\left( z\right) = \frac{1}{2\pi i}{\int }_{C}\... | Proof. The function \( f \) defined by the above integral is analytic on \( D \) by Theorem A3 of \( §6 \) . We have to identify its real part with \( u \) . But the integrand is merely another expression for the convolution of the Poisson kernel with \( u \) . Indeed, the reader will easily verify that if \( z = r{e}^... | Yes |
Theorem 5.5. Let \( u \) be continuous on an open set \( U \) . Suppose that \( u \) satisfies the mean value property locally at every point of \( U \), that is for \( {z}_{0} \in U \) and \( r \) sufficiently small,\n\n\[ u\left( {z}_{0}\right) = {\int }_{0}^{2\pi }u\left( {{z}_{0} + r{e}^{i\theta }}\right) \frac{d\t... | Proof. We first prove that \( u \) satisfies the maximum principle locally. Suppose \( u\left( {z}_{0}\right) \geqq u\left( {{z}_{0} + r{e}^{i\theta }}\right) \) for all \( r \) with \( 0 \leqq r \leqq {r}_{0} \) . Then \( u \) is locally constant at \( {z}_{0} \) . Indeed, suppose that \( u\left( {z}_{1}\right) < u\le... | Yes |
If \( f \) is a function on \( U \), analytic on \( {U}^{ + } \) and \( {U}^{ - } \), and continuous on \( I \), then \( f \) is analytic on \( U \) . | We consider values of \( z \) near \( I \), and especially near some point of \( I \) . Such values lie inside a rectangle, as shown on Fig. 2(a). This rectangle has a boundary \( C = {C}^{ + } + {C}^{ - } \), oriented as shown.\n\n be a continuous function on \( {U}^{ + } \cup I \), harmonic on \( {U}^{ + } \), and equal to 0 on I. Then \( v \) extends to a harmonic function on \( {U}^{ + } \cup I \cup {U}^{ - } \) . | Proof. Define \( v\left( \bar{z}\right) = - v\left( z\right) \) . Then \( v \) is harmonic on \( {U}^{ - } \), because the property of being harmonic can be verified locally, and on a small disc centered at a point \( {z}_{0} \) in \( {U}^{ + }, v \) is the imaginary part of an analytic function \( f \), so \( - v \) i... | Yes |
Proposition 1.3. Let \( f \) be an analytic function on \( U \), real valued on \( I \) . Assume in addition that \( f \) gives an isomorphism of \( {U}^{ + } \) with \( f\left( {U}^{ + }\right) \subset {H}^{ + } \) and an isomorphism of \( {U}^{ - } \) with \( f\left( {U}^{ - }\right) \subset {H}^{ - } \) . Then \( f ... | Proof. We first show that \( f \) is a local isomorphism at each point of \( I \) . After making translations, we may assume without loss of generality that the point of \( I \) is 0, and that \( f\left( 0\right) = 0 \) . Let\n\n\[ f\left( z\right) = c{z}^{m} + \text{ higher terms,}\;\text{ with }\;c \neq 0\text{ and }... | Yes |
(i) Given a function \( g \) on \( V \) which is analytic on \( {V}^{ + } \) and \( {V}^{ - } \), and continuous on \( \gamma \), then \( g \) is analytic on \( V \) . | Proof. Obvious, using successively parts (i) and (ii) of the theorem, applied to the function \( f = g \circ \psi \) . | No |
Theorem 2.2. Let \( f \) be analytic on an open set \( U \) . Let \( \gamma \) be a proper analytic arc which is contained in the boundary of \( U \), and such that \( U \) lies on one side of \( \gamma \) . Assume that \( f \) extends to a continuous function on \( U \cup \gamma \) (i.e. \( U \cup \operatorname{Image}... | Proof. There exist analytic isomorphisms\n\n\[ \varphi : {W}_{1} \rightarrow \text{neighborhood of}\gamma \text{,} \]\n\n\[ \psi : {W}_{2} \rightarrow \text{neighborhood of}\eta \text{,} \]\n\nwhere \( {W}_{1},{W}_{2} \) are open sets as illustrated on Fig. 4, neighborhoods of real intervals \( {I}_{1} \) and \( {I}_{2... | Yes |
Lemma 2.4. Let \( f : U \rightarrow V \) be a proper analytic map. If \( \left\{ {z}_{n}\right\} \) is a sequence in \( U \) approaching the boundary of \( U \), then \( \left\{ {f\left( {z}_{n}\right) }\right\} \) approaches the boundary of \( V \) . | Proof. Given \( {K}^{\prime } \) compact in \( V \), let \( K = {f}^{-1}\left( {K}^{\prime }\right) \) . There is some \( {n}_{0} \) such that for \( n \geqq {n}_{0} \) we have \( {z}_{n} \notin K \), so \( f\left( {z}_{n}\right) \notin {K}^{\prime } \) as desired. | Yes |
Theorem 2.5. Let \( f : U \rightarrow D \) be an analytic isomorphism. Let \( \gamma \) be a proper analytic arc contained in the boundary of \( U \) and such that \( U \) lies on one side of \( \gamma \) . Then \( f \) extends to an analytic isomorphism on \( U \cup \gamma \) . | Proof. In this proof we see the usefulness of dealing with harmonic functions globally. Let \( v\left( z\right) = \log \left| {f\left( z\right) }\right| \) . By Lemma 2.4, it follows that\n\n\[ \mathop{\lim }\limits_{{z \rightarrow \partial U}}v\left( z\right) = 0 \]\n\nTherefore \( v \) extends to a continuous functio... | Yes |
Lemma 2.2. There exists a sequence of compact sets \( {K}_{s}\left( {s = 1,2,\ldots }\right) \) such that \( {K}_{s} \) is contained in the interior of \( {K}_{s + 1} \) and such that the union of all \( {K}_{s} \) is \( U \) . | Proof. Let \( {\bar{D}}_{s} \) be the closed disc of radius \( s \), let \( \bar{U} \) be the closure of \( U \) , and let\n\n\[ \n{K}_{s} = \text{set of points}z \in \bar{U} \cap {\bar{D}}_{s}\text{such that}\operatorname{dist}\left( {z\text{, boundary}U}\right) \geqq 1/s\text{.} \n\]\n\nThen \( {K}_{s} \) is containe... | Yes |
Lemma 3.1. Let \( U \) be an open connected set. Let \( f : U \rightarrow \mathbf{C} \) be analytic and injective. Then \( {f}^{\prime }\left( z\right) \neq 0 \) for all \( z \in U \), and \( f \) is an analytic isomorphism of \( U \) and its image. | This is merely Theorem 6.4 of Chapter II. | No |
Lemma 3.2. Let \( U \) be a connected open set, and let \( \left\{ {f}_{n}\right\} \) be a sequence of injective analytic maps of \( U \) into \( \mathbf{C} \) which converges uniformly on every compact subset of \( U \) . Then the limit function \( f \) is either constant or injective. | Proof. Suppose \( f \) is not injective, so there exist two points \( {z}_{1} \neq {z}_{2} \) in \( U \) such that\n\n\[ f\left( {z}_{1}\right) = f\left( {z}_{2}\right) = \alpha . \]\n\nLet \( {g}_{n} = {f}_{n} - {f}_{n}\left( {z}_{1}\right) \) . Then \( \left\{ {g}_{n}\right\} \) is a sequence which converges uniforml... | Yes |
Theorem 3.3. Let \( f \in \Phi \) be such that \( \left| {{f}^{\prime }\left( 0\right) }\right| \geqq \left| {{g}^{\prime }\left( 0\right) }\right| \) for all \( g \in \Phi \) . Then \( f \) is an analytic isomorphism of \( U \) with the disc. | Proof. The Schwarz lemma for the derivative, Theorem 1.2 of Chapter VII, provides an essential case of the present theorem, and we reduce the proof to that case. All we have to prove is that \( f \) is surjective. Suppose not. Let \( \alpha \in D \) be outside the image of \( f \) . Let \( T \) be an automorphism of th... | Yes |
Theorem 4.1. Let \( U \) be simply connected and bounded, and let\n\n\[ f : U \rightarrow D \]\n\nbe an isomorphism with the disc. If \( \alpha \) is an accessible boundary point of \( U \), then\n\n\[ \mathop{\lim }\limits_{{z \rightarrow \alpha }}f\left( z\right) \]\n\nexists for \( z \in U \), and lies on the unit c... | Proof. Suppose not. Then there exists a sequence \( \left\{ {z}_{n}\right\} \) in \( U \) tending to \( \alpha \), but \( \left\{ {f\left( {z}_{n}\right) }\right\} \) has no limit. We find a curve \( \gamma \) as in the definition of accessibility. | No |
\[ \mathop{\lim }\limits_{{t \rightarrow b}}\left| {f\left( {\gamma \left( t\right) }\right) }\right| = 1 \] | Proof. Suppose not. Given \( \epsilon \) there exists a sequence of increasing numbers \( {s}_{n} \) such that \( \left| {f\left( {\gamma \left( {s}_{n}\right) }\right) }\right| \leqq 1 - \epsilon \), and taking a subsequence if necessary, we may assume \( f\left( {\gamma \left( {s}_{n}\right) }\right) \) converges to ... | Yes |
Theorem 1.2. Let \( P\left( {{T}_{1},{T}_{2}}\right) \) be a polynomial in two variables. Let \( \gamma \) be a curve with beginning point \( {z}_{0} \) and end point \( w \) . Let \( f \) be analytic at \( {z}_{0} \), and suppose that \( f \) has an analytic continuation \( {f}_{\gamma } \) along the curve \( \gamma \... | Proof. This is obvious, because the relation holds in each successive disc \( {D}_{0},{D}_{1},\ldots ,{D}_{m} \) used to carry out the analytic continuation. | No |
Theorem 1.3 (Monodromy Theorem). Let \( U \) be a connected open set. Let \( f \) be analytic at a point \( {z}_{0} \) of \( U \), and let \( \gamma ,\eta \) be two paths from \( {z}_{0} \) to a point \( w \) of \( U \) . Assume:\n\n(i) \( \gamma \) is homotopic to \( \eta \) in \( U \) .\n\n(ii) \( f \) can be continu... | Proof. The proof follows the ideas of Lemma 4.3 and Theorem 5.1 of Chapter III. Let\n\n\[ \psi : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {c, d}\right\rbrack \rightarrow U \]\n\nbe a homotopy so that if we put \( {\gamma }_{u}\left( t\right) = \psi \left( {t, u}\right) \), then \( \gamma = {\gamma }_{c} \) ... | Yes |
Lemma 1.4. For each \( u \in \left\lbrack {c, d}\right\rbrack \), if \( {u}^{\prime } \) is sufficiently close to \( u \), then a continuation of \( f \) along \( {\gamma }_{u} \) is equal to a continuation of \( f \) along \( {\gamma }_{{u}^{\prime }} \) in some neighborhood of \( w \) . | Proof. Given a continuation along a connected sequence of discs or convex open sets\n\n\[ \left( {{f}_{0},{D}_{0}}\right) ,\ldots ,\left( {{f}_{n},{D}_{n}}\right) \]\n\nalong \( {\gamma }_{u} \), connected by the curve along the partition, it is immediately verified that if \( {u}^{\prime } \) is sufficiently close to ... | Yes |
Theorem 1.5. Let \( U \) be a simply connected open set. Let \( {z}_{0} \in U \), and let \( f \) be analytic at \( {z}_{0} \) . Assume that \( f \) can be continued along any path from \( {z}_{0} \) to any point in \( U \) . Let \( {\gamma }_{z} \) be a path from \( {z}_{0} \) to a point \( z \) in \( U \), and let \(... | Proof. Suppose we have shown the independence from the path. Let \( {z}_{1} \) be some point in \( U \) and let \( z \) be a variable point in a disc centered at \( {z}_{1} \) . Then the analytic continuation of \( f \) from \( {z}_{0} \) to \( z \) may be first taken from \( {z}_{0} \) to \( {z}_{1} \) along some path... | No |
Lemma 1.6. Let \( U \) be simply connected. Let \( \gamma ,\eta \) be two paths in \( U \) from a point \( {z}_{0} \) to a point \( {z}_{1} \) . Then there is a homotopy in \( U \) between the two paths, leaving the end points \( {z}_{0},{z}_{1} \) fixed. | Proof. The arguments are routine. Cf. my Undergraduate Analysis, Second Edition, Springer-Verlag, 1997, Chapter XVI, §6, especially Theorem 6.4 and Proposition 6.6. These arguments do not involve complex analysis but merely juggling with homotopies which require being written down, or at least being clearly shown on pi... | No |
Theorem 1.7. Let \( U \) be a simply connected open set. Let \( f \) be analytic at a point \( {z}_{0} \) and assume that \( f \) can be continued along every path from \( {z}_{0} \) to every point in \( U \) . Then the analytic continuation of \( f \) along a path from \( {z}_{0} \) to a point \( w \) is independent o... | Proof. This is a special case of Theorem 1.3, because two paths from \( {z}_{0} \) to \( w \) are homotopic. | Yes |
Theorem 1.1. Let \( f \) be holomorphic on the closed disc of radius \( R \), and assume that \( f\left( 0\right) \neq 0 \) . Let the zeros of \( f \) in the open disc be ordered by increasing absolute value,\n\n\[ \n{z}_{1},{z}_{2},\ldots ,{z}_{N} \n\]\n\neach zero being repeated according to its multiplicity. Then\n\... | Proof. Let\n\n\[ \ng\left( z\right) = \mathop{\prod }\limits_{{n = 1}}^{N}\frac{R\left( {{z}_{n} - z}\right) }{{R}^{2} - {\bar{z}}_{n}z}\;\text{ and }\;F\left( z\right) = \frac{f\left( z\right) }{g\left( z\right) }.\n\]\n\nThen the function \( F \) is holomorphic on the closed disc of radius \( R \), and\n\n\[ \n\left|... | Yes |
Theorem 1.2 (Jensen’s Formula). Let \( f \) be meromorphic and not constant on the closed disc \( {\bar{D}}_{R} \) . Then\n\n\[{\int }_{0}^{2\pi }\log \left| {f\left( {R{e}^{i\theta }}\right) }\right| \frac{d\theta }{2\pi } + \mathop{\sum }\limits_{\substack{{a \in {D}_{R}} \\ {a \neq 0} }}{n}_{f}\left( a\right) \log \... | Proof. Suppose first that \( f \) has no zeros or poles in the closed disc \( \left| z\right| \leqq R \) . Then \( \log f\left( z\right) \) is analytic on this disc, and\n\n\[ \log f\left( 0\right) = \frac{1}{2\pi i}{\int }_{\left| z\right| = R}\frac{\log f\left( z\right) }{z}{dz} = {\int }_{0}^{2\pi }\log f\left( {R{e... | Yes |
Lemma 1.3. If \( 0 < a \leqq R \), then \( {\int }_{0}^{2\pi }\log \left| {{e}^{i\theta } - \frac{a}{R}{e}^{i\varphi }}\right| {d\theta } = 0 \) . | Proof. Suppose first \( a < R \) . Then the function\n\n\[ \frac{\log \left( {1 - \frac{a}{R}z}\right) }{z} \]\n\nis analytic for \( \left| z\right| \leqq 1 \), and from this it is immediate that the desired integral is 0 . Next suppose \( a = R \), so we have to prove that\n\n(*) \n\n\[ {\int }_{0}^{2\pi }\log \left| ... | Yes |
Lemma 2.1. Let \( b \in \mathbf{C} \) . Then\n\n\[{\int }_{0}^{2\pi }\log \left| {b - {e}^{i\theta }}\right| \frac{d\theta }{2\pi } = {\log }^{ + }\left| b\right|\] | Proof. If \( \left| b\right| > 1 \) then \( \log \left| {b - z}\right| \) for \( \left| z\right| < 1 + \varepsilon \) is harmonic, and \( {\log }^{ + }\left| b\right| = \log \left| b\right| \), so the formula is true by the mean value property for harmonic functions. If \( \left| b\right| < 1 \), then\n\n\[{\int }_{0}^... | Yes |
Proposition 2.2 (Cartan). Let \( f \) be an entire function. Then\n\n\[ \n{m}_{f}\left( r\right) = {\int }_{0}^{2\pi }{N}_{f}\left( {{e}^{i\theta }, r}\right) \frac{d\theta }{2\pi } + {\log }^{ + }\left| {f\left( 0\right) }\right| .\n\]\n\nIn particular, \( {m}_{f} \) is an increasing function of \( r \) . | Proof. For each \( \theta \) such that \( f\left( 0\right) \neq {e}^{i\theta } \) we apply Jensen’s formula to\n\nthe function \( f\left( z\right) - {e}^{i\theta } \), to get\n\n\[ \n{N}_{f}\left( {{e}^{i\theta }, r}\right) + \log \left| {f\left( 0\right) - {e}^{i\theta }}\right| = {\int }_{0}^{2\pi }\log \left| {f\lef... | Yes |
Theorem 2.3. Let \( f \) be an entire function. Then for \( r < R \) we have\n\n\[ \n{M}_{f}\left( r\right) \leqq \frac{R + r}{R - r}{m}_{f}\left( R\right) \n\]\n\nand in particular, \( {M}_{f}\left( r\right) \leqq 3{m}_{f}\left( {2r}\right) \) . | Proof. We shall use the theorem only when \( f \) has no zeros, and the proof in this case is slightly shorter. Hence we shall give the proof only in this case. The key step is to show that for \( r < R \) we have the inequality\n\n\[ \n{M}_{f}\left( r\right) \leqq \frac{R + r}{R - r}{m}_{f}\left( R\right) \n\]\n\nSinc... | Yes |
Corollary 2.4. Let \( f \) be an entire function. If \( {m}_{f} \) is bounded for \( r \rightarrow \infty \) then \( f \) is constant. If there exists a constant \( k \) such that \[ {m}_{f}\left( {R}_{j}\right) \leqq k\log {R}_{j} \] for a sequence of numbers \( {R}_{j} \rightarrow \infty \) then \( f \) is a polynomi... | Proof. The first assertion follows from Theorem 2.3 and Liouville's theorem that a bounded entire function is constant. The second assertion is essentially Exercise 5 of Chapter V, \( §1 \), but we give the short proof. Select \( A \) large positive, and let \( R = {Ar} \) . By Cauchy’s theorem, if \( f = \) \( \sum {a... | Yes |
Lemma 2.6. Let \( h \) be an entire function without zeros. Let \( 1 \leqq r < R \) . Then\n\n\[ \n{m}_{{h}^{\prime }/h}\left( r\right) \leqq {\log }^{ + }R + 2{\log }^{ + }\frac{1}{R - r} + 2{\log }^{ + }{m}_{h}\left( R\right) + \text{ a constant. }\n\] | Proof. Note that \( \log h \) is defined as an entire function, and we have the Poisson representation\n\n\[ \n\log h\left( z\right) = {\int }_{0}^{2\pi }\log \left| {h\left( {R{e}^{i\theta }}\right) }\right| \frac{R{e}^{i\theta } + z}{R{e}^{i\theta } - z}\frac{d\theta }{2\pi } + {iK}\n\]\n\nfor some constant \( K \) .... | Yes |
Lemma 2.7. Let \( S \) be a continuous, non-constant, increasing function defined for \( r > 0 \) . Then\n\n\[ S\left( {r + \frac{1}{S\left( r\right) }}\right) < {2S}\left( r\right) \]\n\nfor all \( r > 0 \) except for \( r \) lying in a set of finite measure. | Proof. Let \( E \) be the exceptional set where the stated inequality is false, that is \( S\left( {r + 1/S\left( r\right) }\right) \geqq {2S}\left( r\right) \) . Suppose there is some \( {r}_{1} \in E, S\left( {r}_{1}\right) \neq 0 \) . Let\n\n\[ {r}_{2} = \inf \left\{ {r \in E\text{ such that }r \geqq {r}_{1} + \frac... | Yes |
Theorem 2.8. Let \( f \) be an entire function. Let \( a, b \) be two distinct complex numbers such that \( f\left( z\right) \neq a, b \) for all \( z \) . Then \( f \) is constant. | Proof. Let\n\n\[ L\left( w\right) = \frac{w - a}{b - a}. \]\n\nThen \( L \) carries \( a, b \) to 0,1 . Thus after replacing \( f \) by \( L \circ f \), we may assume without loss of generality that \( a = 0 \) and \( b = 1 \) . Thus we let \( h \) be an entire function which has no zeros and such that \( 1 - h \) also... | No |
Theorem 2.9. Let \( {h}_{1},{h}_{2} \) be entire functions without zeros. If\n\n\[ \n{h}_{1} + {h}_{2} = 1 \n\]\n\nthen \( {h}_{1},{h}_{2} \) are constant. | Differentiating the given relation, we obtain two linear equations\n\n\[ \n{h}_{1} + {h}_{2} = 1 \n\]\n\n\[ \n{h}_{1}^{\prime } + {h}_{2}^{\prime } = 0 \n\]\n\n\nWrite \( {h}_{1}^{\prime } = \left( {{h}_{1}^{\prime }/{h}_{1}}\right) {h}_{1} \) and \( {h}_{2}^{\prime } = \left( {{h}_{2}^{\prime }/{h}_{2}}\right) {h}_{2}... | Yes |
Theorem 3.1 (Borel-Carathéodory). Let \( f \) be holomorphic on a closed disc of radius \( R \), centered at the origin. Let \( \parallel f{\parallel }_{r} = \max \left| {f\left( z\right) }\right| \) for \( \left| z\right| = r < R \) . Then \[ \parallel f{\parallel }_{r} \leqq \frac{2r}{R - r}\mathop{\sup }\limits_{R}\... | Proof. Let \( A = \mathop{\sup }\limits_{R}\operatorname{Re}f \) . Assume first that \( f\left( 0\right) = 0 \) . Then \( A \geqq 0 \) (why?). Let \[ g\left( z\right) = \frac{f\left( z\right) }{z\left( {{2A} - f\left( z\right) }\right) } \] Then \( g \) is holomorphic for \( \left| z\right| \leqq R \) . Furthermore, if... | Yes |
Corollary 3.2. Let \( h \) be an entire function. Let \( \rho > 0 \) . Assume that there exists a number \( C > 0 \) such that for all sufficiently large \( R \) we have\n\n\[ \mathop{\sup }\limits_{R}\operatorname{Re}h \leqq C{R}^{\rho } \]\n\nThen \( h \) is a polynomial of degree \( \leqq \rho \) . | Proof. In the Borel-Carathéodory theorem, use \( R = {2r} \) . Then we have \( \parallel h{\parallel }_{r} \ll {r}^{\rho } \) for \( r \rightarrow \infty \) . Let \( h\left( z\right) = \sum {a}_{n}{z}^{n} \) . By Cauchy’s formula we have \( \left| {a}_{n}\right| \leqq \parallel h{\parallel }_{R}/{R}^{n} \) for all \( R... | Yes |
Corollary 3.3 (Hadamard). Let \( f \) be an entire function with no zeros. Assume that there is a constant \( C \geqq 1 \) such that \( \parallel f{\parallel }_{R} \leqq {C}^{{R}^{\rho }} \) for all \( R \) sufficiently large. Then \( f\left( z\right) = {e}^{h\left( z\right) } \) where \( h \) is a polynomial of degree... | Proof. By Chapter III, \( §6 \) we can define an analytic function \( \log f\left( z\right) = \) \( h\left( z\right) \) such that \( {e}^{h\left( z\right) } = f\left( z\right) \) . The assumption implies that \( \operatorname{Re}h \) satisfies the hypotheses of Corollary 3.2, whence \( h \) is a polynomial, as desired. | No |
Theorem 4.1. Let \( f \) be holomorphic on the closed disc of radius \( R \) . Let \( {z}_{1},\ldots ,{z}_{N} \) be points inside the disc where \( f \) has zeros of multiplicities \( \geqq M \), and assume that these points lie in the disc of radius \( {R}_{1} \) . Assume\n\n\[ \n{R}_{1} \leqq R/2.\n\]\n\nLet \( {R}_{... | Proof. Let \( \left| w\right| = {R}_{2} \) . We estimate the function\n\n\[ \n\frac{f\left( z\right) }{{\left\lbrack \left( z - {z}_{1}\right) \cdots \left( z - {z}_{N}\right) \right\rbrack }^{M}}{\left\lbrack \left( w - {z}_{1}\right) \cdots \left( w - {z}_{N}\right) \right\rbrack }^{M}\n\]\n\non the circle of radius ... | Yes |
Theorem 4.2. Let \( f \) be holomorphic on the closed disc of radius \( R \) . Let \( {z}_{1},\ldots ,{z}_{N} \) be distinct points in the disc of radius \( {R}_{1} \) . Assume\n\n\[ 2{R}_{1} < {R}_{2}\;\text{ and }\;2{R}_{2} < R. \]\n\nLet \( \sigma \) be the minimum of 1, and the distance between any pair of distinct... | An estimate for the derivative of \( f \) can then be obtained from Cauchy's formula\n\n\[ \frac{{D}^{k}f\left( z\right) }{k!} = \frac{1}{2\pi i}{\int }_{\left| \zeta \right| = {R}_{2}}\frac{f\left( \zeta \right) }{{\left( \zeta - z\right) }^{k + 1}}{d\zeta }. \]\n\nfrom which we see that such a derivative is estimated... | No |
Lemma 5.2. Let\n\n\\[ \n{a}_{11}{x}_{1} + \cdots + {a}_{1n}{x}_{n} = 0 \n\\]\n\n\\( \\ldots \\)\n\n\\[ \n{a}_{r1}{x}_{1} + \cdots + {a}_{rn}{x}_{n} = 0 \n\\]\n\nbe a system of linear equations with integer coefficients \\( {a}_{ij} \\), and \\( n > r \\) . Let \\( A \\geqq 1 \\) be a number such that \\( \\left| {a}_{i... | Proof. We view our system of linear equations as a linear equation \\( L\\left( X\\right) = 0 \\), where \\( L \\) is a linear map, \\( L : {\\mathbf{Z}}^{\\left( n\\right) } \\rightarrow {\\mathbf{Z}}^{\\left( r\\right) } \\), determined by the matrix of coefficients. If \\( B \\) is a positive number, we denote by \\... | Yes |
Lemma 5.3. Let \( {f}_{1},\ldots ,{f}_{n} \) be functions such that the derivative \( D = \) \( d/{dz} \) maps the ring \( \mathbf{Q}\left\lbrack {{f}_{1},\ldots ,{f}_{n}}\right\rbrack \) into itself. There exists a number \( {C}_{1} \) having the following property. If \( Q\left( {{T}_{1},\ldots ,{T}_{n}}\right) \) is... | Proof. Let \( {P}_{j}\left( {{T}_{1},\ldots ,{T}_{n}}\right) \) be a polynomial such that\n\n\[ \nD{f}_{j} = {P}_{j}\left( {{f}_{1},\ldots ,{f}_{n}}\right) \n\]\n\nLet \( d \) be the maximum of the degrees of \( {P}_{1},\ldots ,{P}_{n} \) . There exists a \ | Yes |
Theorem 6.1 (Phragmen-Lindelöf). Let \( f \) be continuous on the strip\n\n\[ \n- \pi /2 \leqq \sigma \leqq \pi /2 \n\]\n\nand holomophic on the interior. Suppose \( \left| f\right| \leqq 1 \) on the sides of the strip, and suppose there exists \( 0 < \alpha < 1 \) and \( C > 0 \) such that\n\n\[ \n\left| {f\left( s\ri... | Proof. Let \( \alpha < \beta < 1 \) . For each \( \epsilon > 0 \), define\n\n\[ \n{g}_{\epsilon }\left( s\right) = f\left( s\right) {e}^{-{2\epsilon }\cos \left( {\beta s}\right) }.\n\]\n\nWe have \( 2\operatorname{Re}\left( {\cos {\beta s}}\right) = \left( {{e}^{\beta t} + {e}^{-{\beta t}}}\right) \cos {\beta \sigma }... | Yes |
Theorem 6.2 (Phragmen-Lindelöf, second version). Let \( f \) be continuous on a strip \( \operatorname{Re}\left( s\right) \in \left\lbrack {{\sigma }_{1},{\sigma }_{2}}\right\rbrack \) and holomophic on the interior. Suppose \( \left| f\right| \leqq 1 \) on the sides of the strip, and \( f \) is of finite order on the ... | Proof. By a linear change of variables \( s = {aw} + b = \varphi \left( w\right) \) we see that the given strip corresponds to a strip as in Theorem 6.1. It is immediately verified that \( f \circ \varphi \) satisfies the hypothesis of Theorem 6.1, so \( \left| {f \circ \varphi }\right| \leqq 1 \) , and finally \( \lef... | Yes |
Corollary 6.3. Suppose that \( f \) is of finite order in the strip, and that there is some positive integer \( M \) such that\n\n\[ f\left( {{\sigma }_{1} + {it}}\right) = O\left( {\left| t\right| }^{M}\right) \;\text{ for }\left| t\right| \rightarrow \infty \]\n\nand similarly for \( {\sigma }_{2} \) instead of \( {\... | Proof. Let \( {s}_{0} \) be some point away from the strip. Then the function\n\n\[ g\left( s\right) = f\left( s\right) /{\left( s - {s}_{0}\right) }^{M}. \]\n\n is bounded in the strip and we can apply the Phragment-Lindelöf theorem to conclude the proof. | Yes |
Theorem 6.4 (First Convexity Theorem). Let \( s = \sigma + {it} \) . Let \( f \) be holomorphic and bounded on the strip \( a \leqq \sigma \leqq b \) . For each \( \sigma \) let\n\n\[ \n{M}_{f}\left( \sigma \right) = M\left( \sigma \right) = \mathop{\sup }\limits_{t}\left| {f\left( {\sigma + {it}}\right) }\right| \n\]\... | Proof. The statement is defined to be the inequality\n\n\[ \n\log M\left( \sigma \right) \leqq \frac{b - \sigma }{b - a}\log M\left( a\right) + \frac{\sigma - a}{b - a}\log M\left( b\right) \n\]\n\nExpressed in multiplicative notation, this is equivalent with\n\n\[ \nM{\left( \sigma \right) }^{b - a} \leqq M{\left( a\r... | Yes |
Corollary 6.5 (Hadamard Three Circle Theorem). Let \( f\left( z\right) \) be holomorphic on the annulus \( \alpha \leqq \left| z\right| \leqq \beta \), centered at the origin. Let\n\n\[ M\left( r\right) = \mathop{\sup }\limits_{{\left| z\right| = r}}\left| {f\left( z\right) }\right| \]\n\nThen \( \log M\left( r\right) ... | Proof. Let \( {f}^{ * }\left( s\right) = f\left( {e}^{s}\right) \) . Then \( {f}^{ * } \) is holomorphic and bounded on the strip \( a \leqq \sigma \leqq b \), where \( {e}^{a} = \alpha \) and \( {e}^{b} = \beta \) . We simply apply the theorem, to get the corollary. | No |
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