Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Proposition 3.1 Suppose \( f \) is continuous and has compact support on \( \mathbb{C} \) . Then:\n\n(a) \( u \) given by (6) is also continuous and satisfies (5) in the sense of distributions.\n\n(b) If \( f \) is in the class \( {C}^{k}, k \geq 1 \), then so is \( u \), and \( u \) satisfies (5) in the usual sense.\n... | Proof Note first that\n\n\[ u\left( {z + h}\right) - u\left( z\right) = \frac{1}{\pi }{\int }_{{\mathbb{C}}^{1}}f\left( {z + h - \zeta }\right) - f\left( {z - \zeta }\right) \frac{d\zeta }{\zeta }.\]\n\nand that this tends to zero as \( h \rightarrow 0 \), by the uniform continuity of \( f \) and the fact that the func... | No |
Proposition 3.2 Suppose \( n \geq 2 \) . If \( {f}_{j},1 \leq j \leq n \), are functions of class \( {C}^{k} \) of compact support that satisfy (7), then there exists a function \( u \) of class \( {C}^{k} \) and of compact support that satisfies the inhomogeneous Cauchy-Riemann equations (4). \( {}^{2} \) | Proof Write \( z = \left( {{z}^{\prime },{z}_{n}}\right) \), where \( {z}^{\prime } = \left( {{z}_{1},\ldots ,{z}_{n - 1}}\right) \in {\mathbb{C}}^{n - 1} \) and set\n\n(8)\n\n\[ u\left( z\right) = \frac{1}{\pi }{\int }_{{\mathbb{C}}^{1}}{f}_{n}\left( {{z}^{\prime } \cdot {z}_{n} - \zeta }\right) \frac{{dm}\left( \zeta... | Yes |
Theorem 4.1 Assume \( \Omega \) is a bounded region in \( {\mathbb{C}}^{n} \), whose boundary is of class \( {C}^{3} \), and suppose the complement of \( \bar{\Omega } \) is connected. If \( {F}_{0} \) is a function of class \( {C}^{3} \) on \( \partial \Omega \) that satisfies the tangential Cauchy-Riemann equations, ... | The proof of this theorem is in the same spirit as the previous one, but the details are different. The function \( {F}_{0} \) of class \( {C}^{3}\left( {\partial \Omega }\right) \) can, by definition, be thought of as a function of class \( {C}^{3} \) on the whole space. Now \( {F}_{0} \) satisfies the tangential Cauc... | Yes |
Proposition 5.1 Near any point \( {z}^{0} \in \partial \Omega \) we can introduce holomorphic coordinates \( \left( {{z}_{1},\ldots ,{z}_{n}}\right) \) centered at \( {z}^{0} \) so that\n\n\[ \Omega = \left\{ {\operatorname{Im}\left( {z}_{n}\right) > \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{\lambda }_{j}{\left| {z}_{j... | Proof of the proposition. As in (10), we see that we can introduce complex coordinates (with an affine complex linear change of variables) so that near \( {z}^{0} \) the set \( \Omega \) is given by\n\n\[ \operatorname{Im}\left( {z}_{n}\right) > \varphi \left( {{z}^{\prime },{x}_{n}}\right) \]\n\n\( {}^{5}f\left( z\rig... | Yes |
Corollary 6.2 Suppose the Levi form, as given by (18), has at least one strictly positive eigenvalue for each \( z \in M \) . Under these circumstances, for every \( {z}^{0} \in M \) there is a ball \( {B}^{\prime } \) centered at \( {z}^{0} \) so that whenever \( F \) is holomorphic in \( {\Omega }_{ - } \) and contin... | The theorem we have just proved tells us that when an eigenvalue of the Levi form is positive, the control of the restriction of a holomorphic function to a small piece of the boundary gives us a corresponding control of the function in an interior region. This is a strong hint that for such boundaries a local version ... | No |
Theorem 7.1 Suppose \( M \subset {\mathbb{C}}^{n} \) is a hypersurface of class \( {C}^{2} \) as above. Given a point \( {z}^{0} \in M \), there are open balls \( {B}^{\prime } \) and \( B \), centered at \( {z}^{0} \), with \( {\bar{B}}^{\prime } \subset B \), so that: if \( F \) is a continuous function in \( M \cap ... | Proof. We shall first take \( B \) small enough so that in \( B \), the hypersurface \( M \) has been represented by \( M = \left\{ {{y}_{n} = \varphi \left( {{z}^{\prime },{x}_{n}}\right) }\right\} \) where \( {z}^{0} \) corresponds to the origin. Besides \( \varphi \left( {0,0}\right) = 0 \), we can also suppose that... | Yes |
Corollary 7.3 If \( f \) is a continuous function of compact support, then\n\n\[ \frac{\det \left( {I + A}\right) }{{\epsilon }^{n/2}}{\int }_{{\mathbb{R}}^{n}}{e}^{-\frac{r}{\epsilon }{\left( \left( I + A\right) v\right) }^{2}}f\left( {\xi + v}\right) {dv} \rightarrow f\left( \xi \right) \]\n\nuniformly in \( \xi \) a... | To prove the lemma note that \( \operatorname{Re}\left( {\left( \left( I + A\right) v\right) }^{2}\right) \geq {\left| v\right| }^{2} - \parallel A\parallel {\left| v\right| }^{2} \geq c{\left| v\right| }^{2}. \) with \( c > 0 \), so that the integral in (27) converges. A change of scale reduces the identity to the cas... | Yes |
Theorem 7.5 Suppose that the Levi form (18) has at least one strictly positive eigenvalue for each \( z \in M \) . Then for each \( {z}^{0} \in M \), there is a ball \( {B}^{\prime } \) centered at \( {z}^{0} \) so that whenever \( {F}_{0} \) is a continuous function on \( M \) that satisfies the tangential Cauchy-Riem... | To prove the theorem we first use Theorem 7.1 to find a ball \( {B}_{1} \) centered at \( {z}_{0} \) so that \( {F}_{0} \) can be uniformly approximated (on \( M \cap {B}_{1} \) ) by polynomials \( \left\{ {{p}_{n}\left( z\right) }\right\} \) . Then we invoke the corollary to Theorem 6.1 to find a ball \( {B}^{\prime }... | Yes |
Lemma 8.2 Suppose \( {B}_{1} \) and \( {B}_{2} \) are two open balls in \( {\mathbb{C}}^{n - 1} \), with \( {\bar{B}}_{1} \subset {B}_{2} \). Then, whenever \( f \) is holomorphic in \( {\mathbb{C}}^{n - 1} \n\n\[ \n\mathop{\sup }\limits_{{{z}^{\prime } \in {B}_{1}}}{\left| f\left( {z}^{\prime }\right) \right| }^{2} \l... | Indeed for sufficiently small \( \delta \), whenever \( {z}^{\prime } \in {B}_{1} \) then \( {B}_{\delta }\left( {z}^{\prime }\right) \subset {B}_{2} \), so since \( f \) is harmonic in \( {\mathbb{R}}^{{2n} - 2} \), the mean-value property and the Cauchy-Schwarz inequality gives\n\n\[ \n{\left| f\left( {z}^{\prime }\r... | No |
Lemma 8.6 For \( f \) as above. we have\n\n(39)\n\n\[ f\left( {z}^{\prime }\right) = {\int }_{{\mathbb{C}}^{n - 1}}{K}_{\lambda }\left( {{z}^{\prime },{w}^{\prime }}\right) f\left( {w}^{\prime }\right) {e}^{-{4\pi \lambda }{\left| {w}^{\prime }\right| }^{2}}{dm}\left( {w}^{\prime }\right) \]\n\nwith \( {K}_{\lambda }\l... | Proof In fact, consider first the case when \( {4\lambda } = 1 \), and \( {z}^{\prime } = 0 \) Then (39), which states \( f\left( 0\right) = {\int }_{{\mathbb{C}}^{n - 1}}f\left( {w}^{\prime }\right) {\mathrm{e}}^{-\pi {\left| {u}^{\prime }\right| }^{2}}{dm}\left( {w}^{\prime }\right) \), is a simple consequence of the... | Yes |
Theorem 8.7 Suppose \( U \) is a distribution defined on \( \mathbb{C} \times \mathbb{R} \), so that \( \bar{L}\left( U\right) = f \) in a neighborhood of the origin Then (41) must hold | Proof Assume first that \( U \) has compact support. and \( \bar{L}\left( U\right) = f \) everywhere. Then\n\n\[ C\left( f\right) \left( z\right) = \left\langle {f, S\left( {z.{u}_{2} + i{\left| {w}_{1}\right| }^{2}}\right) }\right\rangle = \left\langle {\bar{L}\left( U\right), S\left( {z,{u}_{2} + i{\left| {w}_{1}\rig... | Yes |
Proposition 1.1 The mapping \( f \mapsto A\left( f\right) \) is bounded from \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) to \( {L}_{k}^{2}\left( {\mathbb{R}}^{d}\right) , \) with \( k = \frac{d - 1}{2} \) . | Proof. The proposition is a consequence of the identity\n\n(3)\n\n\[ \widehat{d\sigma }\left( \xi \right) = {2\pi }{\left| \xi \right| }^{-d/2 + 1}{J}_{d/2 - 1}\left( {{2\pi }\left| \xi \right| }\right) . \]\n\nwhere \( \widehat{d\sigma }\left( \xi \right) = {\int }_{{S}^{d - 1}}{e}^{-{2\pi ix\xi }}{d\sigma }\left( x\r... | Yes |
Proposition 2.1 Suppose \( \left| {\nabla \Phi \left( x\right) }\right| \geq c > 0 \) for all \( x \) in the support of \( \psi \) . Then for every \( N \geq 0 \n\[ \left| {I\left( \lambda \right) }\right| \leq {c}_{N}{\lambda }^{-N},\;\text{ whenever }\lambda > 0. \] | Proof. We consider the following vector field\n\[ L = \frac{1}{i\lambda }\mathop{\sum }\limits_{{k = 1}}^{d}{a}_{k}\frac{\partial }{\partial {x}_{k}} = \frac{1}{i\lambda }\left( {a \cdot \nabla }\right) ,\] \nwith \( \;a = \left( {{a}_{1},\ldots .{a}_{d}}\right) = \frac{\nabla \Phi }{{\left| \nabla \Phi \right| }^{2}}\... | Yes |
Proposition 2.2 In the above situation, \( \left| {{I}_{1}\left( \lambda \right) }\right| \leq c{\lambda }^{-1} \), all \( \lambda > 0 \), with \( c = 3 \) . | Proof. The proof uses the operator \( L \) that occurred in the previous proposition We may assume \( {\Phi }^{\prime } > 0 \) on \( \left\lbrack {a, b}\right\rbrack \), because the case when \( {\Phi }^{\prime } < 0 \) follows by taking complex conjugates. So \( L = \frac{1}{{i\lambda }{\Phi }^{\prime }\left( x\right)... | Yes |
Proposition 2.3 Under the above assumptions, and with \( {I}_{1}\left( \lambda \right) \) given by (7) we have\n\n\[ \left| {{I}_{1}\left( \lambda \right) }\right| \leq {c}^{\prime }{\lambda }^{-1/2}\;\text{ for all }\lambda > 0,\text{ with }{c}^{\prime } = 8. \] | Proof. We may assume that \( {\Phi }^{\prime \prime }\left( x\right) \geq 1 \) throughout the interval, because the case \( {\Phi }^{\prime \prime }\left( x\right) \leq - 1 \) follows from this by taking complex conjugates. Now \( {\Phi }^{\prime \prime }\left( x\right) \geq 1 \) implies that \( {\Phi }^{\prime }\left(... | Yes |
Corollary 2.4 Assume \( \Phi \) satisfies the hypotheses of Proposition 2.3. Then\n\n\[ \left| {{\int }_{a}^{b}{e}^{{i\lambda \Phi }\left( s\right) }\psi \left( x\right) {dx}}\right| \leq {c}_{w}{\lambda }^{-1/2} \]\n\nwhere \( {c}_{\psi } = 8\left( {{\int }_{a}^{b}\left| {{\psi }^{\prime }\left( x\right) }\right| {dx}... | Proof. Let \( J\left( x\right) = {\int }_{a}^{x}{e}^{{i\lambda \Phi }\left( u\right) }{du} \) We integrate by parts, using \( J\left( a\right) = \) 0. Then\n\n\[ {\int }_{a}^{b}{e}^{{i\lambda \Phi }\left( x\right) }\psi \left( x\right) {dx} = - {\int }_{a}^{b}J\left( x\right) \frac{d\psi }{dx}{dx} + J\left( b\right) \p... | Yes |
Corollary 3.2 If \( M \) has at least \( m \) non-vanishing principal curvatures at each point of the support of \( {d\mu } \) . then\n\n\[ \left| {\widehat{d\mu }\left( \xi \right) }\right| = O\left( {\left| \xi \right| }^{-m/2}\right) \;\text{ as }\left| \xi \right| \rightarrow \infty . \] | First some preliminary remarks. We can assume that the support of \( \psi \) is centered in a sufficiently small ball (so that in particular the representation (18) of \( M \) holds in it), because we can always write a given \( \psi \) as a finite sum of \( {\psi }_{j} \) of that type. Next, all our estimates can be m... | No |
Corollary 3.3 If \( M = \partial \Omega \) has non-vanishing Gauss curvature at each point, then\n\n\[ \n{\widehat{\chi }}_{\Omega }\left( \xi \right) = O\left( {\left| \xi \right| }^{-\frac{d + 1}{2}}\right) ,\;\text{ as }\left| \xi \right| \rightarrow \infty .\n\] | Proof. Using an appropriate partition of unity we can write\n\n\[ \n{\chi }_{\Omega } = \mathop{\sum }\limits_{{j = 0}}^{N}{\psi }_{j}{\chi }_{\Omega }\n\]\n\nwith each \( {\psi }_{j} \) a \( {C}^{\infty } \) function of compact support; \( {\psi }_{0} \) is supported in the interior of \( \Omega \), while each \( {\ps... | Yes |
Corollary 4.3 If we only assume that \( M \) has at least \( m \) non-vanishing principal curvatures, then the same conclusions hold with \( k = m/2 \), and \( p = \frac{m + 2}{m + 1}, q = m + 2. | The proof of part (a) in the theorem is the same as that for the sphere once we invoke the decay (21), which implies that \( {\left( 1 + {\left| \xi \right| }^{2}\right) }^{k/2}\widehat{d\mu }\left( \xi \right) \) is bounded. Hence\n\n\[ \parallel A\left( f\right) {\parallel }_{{L}_{k}^{2}} = \parallel {\left( 1 + {\le... | Yes |
Proposition 4.4 With the above assumptions,\n\n\[ \n{\begin{Vmatrix}{T}_{c}\end{Vmatrix}}_{{L}^{q}} \leq M\parallel f{\parallel }_{{L}^{p}} \n\]\n\nfor any \( c \) with \( a \leq c \leq b \), where \( c = \left( {1 - \theta }\right) a + {\theta b} \) and \( 0 \leq \theta \leq 1 \) ; and\n\n\[ \n\frac{1}{p} = \frac{1 - ... | Once we have formulated this result, we in fact observe that we can prove it by essentially the same argument as in Section 2 in Chapter 2.\n\nWe write \( s = a\left( {1 - z}\right) + {bz} \), so \( z = \frac{s - a}{b - a} \) . and the strip \( S \) is thereby transformed into the strip \( 0 \leq \operatorname{Re}\left... | Yes |
Lemma 4.6 \( {I}_{s}\left( \rho \right) \) initially given above for \( \operatorname{Re}\left( s\right) > 0 \), has an analytic continuation into the half-space \( \operatorname{Re}\left( s\right) > - N - 1 \) . | Proof. Write \( s\left( {s + 1}\right) \cdots \left( {s + N}\right) {u}^{s - 1} = {\left( \frac{d}{du}\right) }^{N + 1}{u}^{s + N} \) . Then an \( \left( {N + 1}\right) \) -fold integration by parts yields\n\n\[ \n{I}_{s}\left( \rho \right) = {\left( -1\right) }^{N + 1}{\int }_{0}^{\infty }{u}^{s + N}{\left( \frac{d}{d... | Yes |
Proposition 5.1 Suppose \( f \in {L}^{p}\left( {\mathbb{R}}^{d}\right) \) is a radial function. Then \( \widehat{f} \) is continuous for \( \xi \neq 0 \) whenever \( 1 \leq p < {2d}/\left( {d + 1}\right) \) . Note the sequence of exponents \( \frac{2d}{\left( d + 1\right) }:\;1.\;\frac{4}{3},\;\frac{3}{2}.\;\frac{8}{5}... | Proof. Suppose \( f\left( x\right) = {f}_{0}\left( \left| x\right| \right) \) . Then \( \widehat{f}\left( \xi \right) = F\left( \left| \xi \right| \right) \) with \( F \) defined by (4). namely.\n\n(29)\n\n\[ F\left( \rho \right) = {2\pi }{\rho }^{-d/2 + 1}{\int }_{0}^{\infty }{J}_{d/2 - 1}\left( {2\pi \rho r}\right) {... | Yes |
Theorem 5.2 Suppose \( M \) has non-zero Gauss curvature at each point of the support of \( {d\mu } \) . Then the restriction inequality (31) holds for \( q = 2 \) and \( p = \frac{{2d} + 2}{d + 3} \) . | The proof starts with several quick observations. Let \( \mathcal{R} \) denote the restriction operator\n\n\[ \mathcal{R}\left( f\right) = {\left. \widehat{f}\left( \xi \right) \right| }_{M} = {\left. {\int }_{{\mathbb{R}}^{d}}{e}^{-{2\pi ix\xi }}f\left( x\right) dx\right| }_{M}. \]\n\nwhich is initially defined to map... | No |
Corollary 5.4 Under the assumptions of the theorem, the restriction inequality (31) holds for \( 1 \leq p \leq \frac{{2d} + 2}{d + 3} \) and \( q \leq \left( \frac{d - 1}{d + 1}\right) {p}^{\prime } \) . | This follows by combining the critical case \( \;p = \frac{{2d} + 2}{d + 3},\;q \leq 2\;( \) a consequence of the theorem and Hölder's inequality) with the trivial case \( p = 1.q = \infty \) via the Riesz interpolation theorem. | Yes |
Proposition 6.1 For each \( t \) :\n\n(i) \( {e}^{{it}\bigtriangleup } \) maps \( \mathcal{S} \) to \( \mathcal{S} \) .\n\n(ii) If we set \( u\left( {x, t}\right) = {e}^{{it}\bigtriangleup }\left( f\right) \left( x\right) \), with \( f \in \mathcal{S} \), then \( u \) is a \( {C}^{\infty } \) function of \( \left( {x, ... | Proof. That \( {e}^{{it}\bigtriangleup } \) maps \( \mathcal{S} \) to \( \mathcal{S} \) is clear because the multiplier \( {e}^{-{it4}{\pi }^{2}{\left| \xi \right| }^{2}} \) has the property that each derivative in \( \xi \) is of at most polynomial increase. Next, the Fourier inversion formula gives\n\n\[ u\left( {x, ... | Yes |
Proposition 6.2 For each \( t \) :\n\n(i) The operator \( {e}^{{it}\bigtriangleup } \) is unitary on \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) .\n\n(ii) For every \( f \), the mapping \( t \mapsto {e}^{{it}\bigtriangleup }\left( f\right) \) is continuous in the \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) norm.\n\n(ii... | Proof. Conclusion (i) is immediate from Plancherel's theorem, since the multiplier \( {e}^{-{it4}{\pi }^{2}{\left| \xi \right| }^{2}} \) has absolute value one. Now if \( \widehat{f} \in {L}^{2}\left( {\mathbb{R}}^{d}\right) \) , then clearly \( {e}^{-{it4}{\pi }^{2}{\left| \xi \right| }^{2}}\widehat{f}\left( \xi \righ... | Yes |
Theorem 6.4 The solution \( {e}^{t{\left( \frac{d}{d\bar{x}}\right) }^{3}}\left( f\right) \) satisfies\n\n\[ \parallel u{\parallel }_{{L}^{q}\left( {\mathbb{R}}^{2}\right) } \leq c\parallel f{\parallel }_{{L}^{2}\left( \mathbb{R}\right) }.\;\text{ with }q = 8. \] | The proof of this is result is parallel with that of the previous theorem and reduces to a restriction theorem on \( {\mathbb{R}}^{2} \) for the cubic curve\n\n\[ \Gamma = \left\{ {\left( {{\xi }_{1},{\xi }_{2}}\right) : {\xi }_{2} = - 4{\pi }^{2}{\xi }_{1}^{3}}\right\} \]\n\nAccording to Corollary 5.5, what is needed ... | Yes |
Lemma 6.5 Let \( I\left( \xi \right) = {\int }_{\mathfrak{T}}{e}^{{2\pi i}\left( {{\xi }_{1}t + {\xi }_{2}{t}^{3}}\right) }\psi \left( t\right) {dt} \), where \( \psi \) is a \( {C}^{\infty } \) function of compact support. Then\n\n\[ I\left( \xi \right) = O\left( {\left| \xi \right| }^{-1/3}\right) ,\;\text{ as }\left... | Proof. First note that \( I\left( \xi \right) = O\left( {\left| {\xi }_{2}\right| }^{-1/3}\right) \) . In fact\n\n\[ I\left( \xi \right) = {\int }_{\left| t\right| \leq {\left| {\xi }_{2}\right| }^{-1/3}} + {\int }_{\left| t\right| > {\left| {\xi }_{2}\right| }^{-1/3}}. \]\n\nThe first integral is obviously \( O\left( ... | Yes |
Proposition 6.6 Suppose \( F \) is a \( {C}^{\infty } \) function on \( {\mathbb{R}}^{d} \times \mathbb{R} \) of compact support. Then \( S\left( F\right) \) is a \( {C}^{\infty } \) function that satisfies (43) and (44). | Proof. Write \( F = {e}^{{it}\bigtriangleup }G\left( {\cdot, t}\right) \) with \( G\left( {x, t}\right) = i{\int }_{0}^{l}{e}^{-{is}\bigtriangleup }F\left( {\cdot, s}\right) {ds} \) . Now \( F\left( {\cdot, s}\right) \) is in the Schwartz space \( \mathcal{S}\left( {\mathbb{R}}^{d}\right) \) for each \( s \) and depend... | Yes |
Proposition 6.8 If \( F \in {L}^{p}\left( {{\mathbb{R}}^{d} \times \mathbb{R}}\right) \) then \( S\left( F\right) \) can be corrected (that is, redefined on a set of measure zero) so that for each \( t, S\left( F\right) \left( {\cdot .t}\right) \) belongs to \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) and, moreover, th... | This is based on the inequality\n\n(52)\n\n\[ \n{\begin{Vmatrix}{\int }_{\alpha }^{\beta }{e}^{-{is}\bigtriangleup }F\left( \cdot, s\right) ds\end{Vmatrix}}_{{L}^{2}\left( {\mathbb{R}}^{d}\right) } \leq c\parallel F{\parallel }_{{L}^{p}\left( {{\mathbb{R}}^{d} \times \mathbb{R}}\right) }, \]\n\nwith \( c \) independent... | Yes |
Proposition 7.4 Assume that\n\n\[ \n\begin{Vmatrix}{{T}_{k}{T}_{j}^{ * }}\end{Vmatrix} \leq {a}^{2}\left( {k - j}\right) \;\text{ and }\;\begin{Vmatrix}{{T}_{k}^{ * }{T}_{j}}\end{Vmatrix} \leq {a}^{2}\left( {k - j}\right) .\n\]\n\nThen for every \( r \) ,\n\n(72)\n\n\[ \n\begin{Vmatrix}{\mathop{\sum }\limits_{{k = 0}}^... | Proof. We write \( T = \mathop{\sum }\limits_{{k = 0}}^{r}{T}_{k} \) and recall that \( \parallel T{\parallel }^{2} = \begin{Vmatrix}{T{T}^{ * }}\end{Vmatrix} \) . Since \( T{T}^{ * } \) is self-adjoint we may use this identity repeatedly to obtain \( \parallel T{\parallel }^{2n} = \) \( \begin{Vmatrix}{\left( T{T}^{ *... | Yes |
Proposition 8.1 \( \mathop{\sum }\limits_{{k = 1}}^{\mu }{r}_{2}\left( k\right) = {\pi \mu } + O\left( {\mu }^{1/2}\right) \), as \( \mu \rightarrow \infty \) . | The proof depends on the realization that \( \mathop{\sum }\limits_{{k = 0}}^{\mu }{r}_{2}\left( k\right) \) represents the number of lattice points in the disc of radius \( R \) with \( {R}^{2} = \mu \) . In fact, with \( {\mathbb{Z}}^{2} \) denoting the lattice points in \( {\mathbb{R}}^{2} \), that is, the points in... | Yes |
Proposition 8.2 Suppose \( f \) belongs to the Schwartz space \( \mathcal{S}\left( {\mathbb{R}}^{d}\right) \) . Then\n\n\[\n\mathop{\sum }\limits_{{n \in {\mathbb{Z}}^{d}}}f\left( n\right) = \mathop{\sum }\limits_{{n \in {\mathbb{Z}}^{d}}}\widehat{f}\left( n\right)\n\]\nHere \( {\mathbb{Z}}^{d} \) denotes the collectio... | For the proof consider two sums\n\n\[\n\mathop{\sum }\limits_{{n \in {\mathbb{Z}}^{d}}}f\left( {x + n}\right) \;\text{ and }\;\mathop{\sum }\limits_{{n \in {\mathbb{Z}}^{d}}}\widehat{f}\left( n\right) {e}^{2\pi inx}.\n\]\nBoth are rapidly converging series (since \( f \) and \( \widehat{f} \) are in \( \mathcal{S}\left... | Yes |
Theorem 8.3 \( N\left( R\right) = \pi {R}^{2} + O\left( {R}^{2/3}\right) \), as \( R \rightarrow \infty \) . | Proof. We replace the characteristic function \( {\chi }_{R} \) by a regularized version as follows. We fix a non-negative \ | No |
Corollary 8.7 The conclusions for \( {\mathfrak{J}}_{a, b}^{ - } \) are the same as those for \( {\mathfrak{J}}_{a, b}^{ + } \) stated in Proposition 8.6, except that (i) should be modified to read that uniformly in \( a, b \) , \[ \text{(i’)}{\mathfrak{J}}_{a, b}^{ - } = O\left( {\left| \lambda \right| }^{-N}\right) \... | The only change occurs in the treatment of \( {II} \), namely \( \int {e}^{{i\lambda \Phi }\left( u\right) }\alpha \left( u\right) \;\frac{du}{u}, \) where now \( \Phi \left( u\right) = u - 1/u \) . In this case \( {\Phi }^{\prime }\left( u\right) = 1 + 1/{u}^{2} > 1 \), and there is no critical point. So Proposition 2... | Yes |
Theorem 8.9 Let \( \widehat{f} \) be the Fourier transform of \( f\left( {x, y}\right) = {f}_{0}\left( {xy}\right) \) . Then \( \widehat{f} \) is a continuous function where \( {\xi \eta } \neq 0 \) . It is given by\n\n\[ \widehat{f}\left( {\xi .\eta }\right) = 2{\int }_{0}^{\infty }{\mathfrak{J}}^{ + }\left( {-{2\pi }... | Proof. We approximate \( f \) by \( {f}_{\epsilon } \) . with \( {f}_{\epsilon }\left( {x, y}\right) = {f}_{0}\left( {xy}\right) {\eta }_{\epsilon }\left( x\right) {\eta }_{\epsilon }\left( y\right) \) . Then each \( {f}_{\epsilon } \) is a \( {C}^{\infty } \) function of compact support, and clearly \( {f}_{\epsilon }... | Yes |
Corollary 8.10 The Fourier transforms \( {\widehat{f}}_{\epsilon } \) and \( \widehat{f} \) satisfy the following estimate, uniformly in \( \epsilon \) :\n\n(94)\n\n\[ \left| {{\widehat{f}}_{\epsilon }\left( {\xi ,\eta }\right) }\right| \leq {A}_{N}{\left| \xi \eta \right| }^{-N}\;\text{ when }\left| {\xi \eta }\right|... | This is a consequence of the asymptotic behavior of \( {\mathfrak{J}}^{ \pm }\left( \lambda \right) \) for \( \lambda \) as given in Proposition 86 and its corollary together with the fact that \( {\int }_{0}^{\infty }{e}^{-{4\pi i\rho }{\left| \xi \eta \right| }^{1/2}}{f}_{0}\left( {\rho }^{2}\right) {\rho d\rho } \) ... | Yes |
That weakly holomorphic functions (with values in a complex Banach space) are strongly holomorphic | was proved by N. Dunford in Trans. Amer. Math. Soc., vol. 44, pp. 304-356, 1938. | Yes |
The existence of fundamental solutions (Theorem 8.5) | The existence of fundamental solutions (Theorem 8.5) was established independently by L. Ehrenpreis (Amer. J. Math., vol. 76, pp. 883-903, 1954) and by B. Malgrange in his thesis (Ann. Inst. Fourier, vol. 6, pp. 271-355, 1955-1956). Lemma 8.3 is Malgrange's. He proves it for Fourier transforms \( f \) of test functions... | Yes |
Theorem 11.9. When \( A \) has no unit, then \( \Delta \) is locally compact (but not compact) and \( \widehat{A} \subset {C}_{0}\left( \Delta \right) \) ; the origin of \( {A}^{ * } \) is then in the closure of \( \Delta \) . | See [16], pp. 52-53. | No |
Theorem 11.18 was proved by Gelfand and Naimark in Mat. Sbornik, vol. 12, pp. 197-213, 1943. In the same paper they also proved that every \( {B}^{ * } \) -algebra \( A \) (commutative or not) is isometrically *-isomorphic to an algebra of bounded operators on some Hilbert space (Theorem 12.41), if \( e + {x}^{ * }x \)... | That this additional hypothesis is redundant was proved 15 years later by I. Kaplansky \( \lbrack \left( f\right) \) of Theorem 11.28]. See [21], p. 248, for references to the rather tangled history of this theorem. | No |
Theorem 12.38 | Theorem 12.38 was proved by P. R. Halmos, G. Lumer, and J. Schäffer, in Proc. Amer. Math. Soc., vol. 4, pp. 142-149, 1953. | Yes |
Theorem 13.6 was first proved by A. Wintner, Phys. Rev., vol. 71, pp. 738-739, 1947. The more algebraic proof of the text is H. Wielandt's, Math. Ann., vol. 121, p. 21, 1949. | It was generalized by D. C. Kleinecke (Proc. Amer. Math. Soc., vol. 8, pp. 535-536, 1957), to yield the following theorem about derivations: If \( D \) is a continuous linear operator in a Banach algebra \( A \) such that \( D\left( {xy}\right) = {xDy} + \left( {Dx}\right) y \) for all \( x, y \in A \), then the spectr... | Yes |
Lemma 2 Suppose that \( \gamma = \sum {c}_{n}/{3}^{n} \) with \( {c}_{n} = 0,2 \) or -2. If \( \left| \gamma \right| \leq \) \( 1/{3}^{N + 1} \), then \( {c}_{k} = 0 \) for \( 1 \leq k \leq N - 1 \) . | In Exercise 11, §2, we proved the analogue results for decimal expansions. The proofs of the above lemmas are an \( \epsilon \) -variation of the proofs given in Exercise 11, §2. | No |
Theorem 1 Let \( f \) be a differentiable, non-negative decreasing function and let \( h \) be a continuous function. Suppose that there exists \( M > 0 \) such that\n\n\[ \n\\left| {{\\int }_{A}^{B}h\\left( x\\right) {dx}}\\right| \\leq M \n\]\n\nfor all \( A \\leq B \) . Then\n\n\[ \n\\left| {{\\int }_{A}^{B}f\\left(... | The proof goes as follows. Let \( H\\left( x\\right) = {\\int }_{A}^{x}h\\left( t\\right) {dt} \) . Integrating by parts we\nobtain\n\[ \n{\\int }_{A}^{B}f\\left( x\\right) g\\left( x\\right) {dx} = {\\left\\lbrack f\\left( x\\right) H\\left( x\\right) \\right\\rbrack }_{A}^{B} - {\\int }_{A}^{B}{f}^{\\prime }\\left( x... | Yes |
Proposition 1.4 If \( {\Omega }_{1} \supset {\Omega }_{2} \supset \cdots \supset {\Omega }_{n} \supset \cdots \) is a sequence of non-empty compact sets in \( \mathbb{C} \) with the property that\n\n\[ \operatorname{diam}\left( {\Omega }_{n}\right) \rightarrow 0\;\text{ as }n \rightarrow \infty ,\]\n\nthen there exists... | Proof. Choose a point \( {z}_{n} \) in each \( {\Omega }_{n} \) . The condition \( \operatorname{diam}\left( {\Omega }_{n}\right) \rightarrow 0 \) says precisely that \( \left\{ {z}_{n}\right\} \) is a Cauchy sequence, therefore this sequence converges to a limit that we call \( w \) . Since each set \( {\Omega }_{n} \... | Yes |
Theorem 2.1 A continuous function on a compact set \( \Omega \) is bounded and attains a maximum and minimum on \( \Omega \) . | This is of course analogous to the situation of functions of a real variable, and we shall not repeat the simple proof here. | No |
Proposition 2.3 If \( f \) is holomorphic at \( {z}_{0} \), then\n\n\[ \frac{\partial f}{\partial \bar{z}}\left( {z}_{0}\right) = 0\;\text{ and }\;{f}^{\prime }\left( {z}_{0}\right) = \frac{\partial f}{\partial z}\left( {z}_{0}\right) = 2\frac{\partial u}{\partial z}\left( {z}_{0}\right) . \]\n\nAlso, if we write \( F\... | Proof. Taking real and imaginary parts, it is easy to see that the Cauchy-Riemann equations are equivalent to \( \partial f/\partial \bar{z} = 0 \) . Moreover, by our earlier observation\n\n\[ {f}^{\prime }\left( {z}_{0}\right) = \frac{1}{2}\left( {\frac{\partial f}{\partial x}\left( {z}_{0}\right) + \frac{1}{i}\frac{\... | Yes |
Theorem 2.4 Suppose \( f = u + {iv} \) is a complex-valued function defined on an open set \( \Omega \) . If \( u \) and \( v \) are continuously differentiable and satisfy the Cauchy-Riemann equations on \( \Omega \), then \( f \) is holomorphic on \( \Omega \) and \( {f}^{\prime }\left( z\right) = \partial f/\partial... | Proof. Write\n\n\[ u\left( {x + {h}_{1}, y + {h}_{2}}\right) - u\left( {x, y}\right) = \frac{\partial u}{\partial x}{h}_{1} + \frac{\partial u}{\partial y}{h}_{2} + \left| h\right| {\psi }_{1}\left( h\right) \]\n\nand\n\n\[ v\left( {x + {h}_{1}, y + {h}_{2}}\right) - v\left( {x, y}\right) = \frac{\partial v}{\partial x... | Yes |
Theorem 2.5 Given a power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n} \), there exists \( 0 \leq R \leq \infty \) such that:\n\n(i) If \( \left| z\right| < R \) the series converges absolutely.\n\n(ii) If \( \left| z\right| > R \) the series diverges.\n\nMoreover, if we use the convention that \( ... | Proof. Let \( L = 1/R \) where \( R \) is defined by the formula in the statement of the theorem, and suppose that \( L \neq 0,\infty \) . (These two easy cases are left as an exercise.) If \( \left| z\right| < R \), choose \( \epsilon > 0 \) so small that\n\n\[ \left( {L + \epsilon }\right) \left| z\right| = r < 1 \]\... | No |
Corollary 2.7 A power series is infinitely complex differentiable in its disc of convergence, and the higher derivatives are also power series obtained by termwise differentiation. | We have so far dealt only with power series centered at the origin. More generally, a power series centered at \( {z}_{0} \in \mathbb{C} \) is an expression of the form\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{\left( z - {z}_{0}\right) }^{n}. \]\n\nThe disc of convergence of \( f \) is ... | Yes |
Proposition 3.1 Integration of continuous functions over curves satisfies the following properties:\n\n(i) It is linear, that is, if \( \alpha ,\beta \in \mathbb{C} \), then\n\n\[ \n{\int }_{\gamma }\left( {{\alpha f}\left( z\right) + {\beta g}\left( z\right) }\right) {dz} = \alpha {\int }_{\gamma }f\left( z\right) {dz... | Proof. The first property follows from the definition and the linearity of the Riemann integral. The second property is left as an exercise. For the third, note that\n\n\[ \n\left| {{\int }_{\gamma }f\left( z\right) {dz}}\right| \leq \mathop{\sup }\limits_{{t \in \left\lbrack {a, b}\right\rbrack }}\left| {f\left( {z\le... | No |
Theorem 3.2 If a continuous function \( f \) has a primitive \( F \) in \( \Omega \), and \( \gamma \) is a curve in \( \Omega \) that begins at \( {w}_{1} \) and ends at \( {w}_{2} \), then\n\n\[ \n{\int }_{\gamma }f\left( z\right) {dz} = F\left( {w}_{2}\right) - F\left( {w}_{1}\right) \n\] | Proof. If \( \gamma \) is smooth, the proof is a simple application of the chain rule and the fundamental theorem of calculus. Indeed, if \( z\left( t\right) : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) is a parametrization for \( \gamma \), then \( z\left( a\right) = {w}_{1} \) and \( z\left( b\right) ... | Yes |
Corollary 3.3 If \( \gamma \) is a closed curve in an open set \( \Omega \), and \( f \) is continuous and has a primitive in \( \Omega \), then\n\n\[{\int }_{\gamma }f\left( z\right) {dz} = 0\] | This is immediate since the end-points of a closed curve coincide. | No |
Corollary 3.4 If \( f \) is holomorphic in a region \( \Omega \) and \( {f}^{\prime } = 0 \), then \( f \) is constant. | Proof. Fix a point \( {w}_{0} \in \Omega \) . It suffices to show that \( f\left( w\right) = f\left( {w}_{0}\right) \) for all \( w \in \Omega \) .\n\nSince \( \Omega \) is connected, for any \( w \in \Omega \), there exists a curve \( \gamma \) which joins \( {w}_{0} \) to \( w \) . Since \( f \) is clearly a primitiv... | Yes |
Corollary 1.2 If \( f \) is holomorphic in an open set \( \Omega \) that contains a rectangle \( R \) and its interior, then\n\n\[ \n{\int }_{R}f\left( z\right) {dz} = 0 \n\] | This is immediate since we first choose an orientation as in Figure 2 and note that\n\n\[ \n{\int }_{R}f\left( z\right) {dz} = {\int }_{{T}_{1}}f\left( z\right) {dz} + {\int }_{{T}_{2}}f\left( z\right) {dz}. \n\] | Yes |
Theorem 2.2 (Cauchy's theorem for a disc) If \( f \) is holomorphic in a disc, then\n\n\[{\int }_{\gamma }f\left( z\right) {dz} = 0\]\n\nfor any closed curve \( \gamma \) in that disc. | Proof. Since \( f \) has a primitive, we can apply Corollary 3.3 of Chapter 1. | No |
Corollary 2.3 Suppose \( f \) is holomorphic in an open set containing the circle \( C \) and its interior. Then\n\n\[{\int }_{C}f\left( z\right) {dz} = 0\] | Proof. Let \( D \) be the disc with boundary circle \( C \) . Then there exists a slightly larger disc \( {D}^{\prime } \) which contains \( D \) and so that \( f \) is holomorphic on \( {D}^{\prime } \) . We may now apply Cauchy’s theorem in \( {D}^{\prime } \) to conclude that \( {\int }_{C}f\left( z\right) {dz} = 0.... | Yes |
Theorem 4.1 Suppose \( f \) is holomorphic in an open set that contains the closure of a disc D. If \( C \) denotes the boundary circle of this disc with the positive orientation, then\n\n\[ f\left( z\right) = \frac{1}{2\pi i}{\int }_{C}\frac{f\left( \zeta \right) }{\zeta - z}{d\zeta }\;\text{ for any point }z \in D. \... | Proof. Fix \( z \in D \) and consider the \ | No |
If \( f \) is holomorphic in an open set \( \Omega \), then \( f \) has infinitely many complex derivatives in \( \Omega \) . Moreover, if \( C \subset \Omega \) is a circle whose interior is also contained in \( \Omega \), then\n\n\[ \n{f}^{\left( n\right) }\left( z\right) = \frac{n!}{2\pi i}{\int }_{C}\frac{f\left( \... | The proof is by induction on \( n \), the case \( n = 0 \) being simply the Cauchy integral formula. Suppose that \( f \) has up to \( n - 1 \) complex derivatives and that\n\n\[ \n{f}^{\left( n - 1\right) }\left( z\right) = \frac{\left( {n - 1}\right) !}{2\pi i}{\int }_{C}\frac{f\left( \zeta \right) }{{\left( \zeta - ... | Yes |
Corollary 4.3 (Cauchy inequalities) If \( f \) is holomorphic in an open set that contains the closure of a disc \( D \) centered at \( {z}_{0} \) and of radius \( R \) , then\n\n\[ \left| {{f}^{\left( n\right) }\left( {z}_{0}\right) }\right| \leq \frac{n!\parallel f{\parallel }_{C}}{{R}^{n}} \]\n\nwhere \( \parallel f... | Proof. Applying the Cauchy integral formula for \( {f}^{\left( n\right) }\left( {z}_{0}\right) \), we obtain\n\n\[ \left| {{f}^{\left( n\right) }\left( {z}_{0}\right) }\right| = \left| {\frac{n!}{2\pi i}{\int }_{C}\frac{f\left( \zeta \right) }{{\left( \zeta - {z}_{0}\right) }^{n + 1}}{d\zeta }}\right| \]\n\n\[ = \frac{... | Yes |
Theorem 4.4 Suppose \( f \) is holomorphic in an open set \( \Omega \) . If \( D \) is a disc centered at \( {z}_{0} \) and whose closure is contained in \( \Omega \), then \( f \) has a power series expansion at \( {z}_{0} \) for all \( z \in D \), and the coefficients are given by \( {a}_{n} = \frac{{f}^{\left( n\rig... | Proof. Fix \( z \in D \) . By the Cauchy integral formula, we have \[ f\left( z\right) = \frac{1}{2\pi i}{\int }_{C}\frac{f\left( \zeta \right) }{\zeta - z}{d\zeta } \] where \( C \) denotes the boundary of the disc and \( z \in D \) . The idea is to write \[ \frac{1}{\zeta - z} = \frac{1}{\zeta - {z}_{0} - \left( {z -... | Yes |
Corollary 4.5 (Liouville's theorem) If \( f \) is entire and bounded, then \( f \) is constant. | Proof. It suffices to prove that \( {f}^{\prime } = 0 \), since \( \mathbb{C} \) is connected, and we may then apply Corollary 3.4 in Chapter 1.\n\nFor each \( {z}_{0} \in \mathbb{C} \) and all \( R > 0 \), the Cauchy inequalities yield\n\n\[ \left| {{f}^{\prime }\left( {z}_{0}\right) }\right| \leq \frac{B}{R} \]\n\nwh... | Yes |
Corollary 4.6 Every non-constant polynomial \( P\left( z\right) = {a}_{n}{z}^{n} + \cdots + {a}_{0} \) with complex coefficients has a root in \( \mathbb{C} \) . | Proof. If \( P \) has no roots, then \( 1/P\left( z\right) \) is a bounded holomorphic function. To see this, we can of course assume that \( {a}_{n} \neq 0 \), and write\n\n\[ \frac{P\left( z\right) }{{z}^{n}} = {a}_{n} + \left( {\frac{{a}_{n - 1}}{z} + \cdots + \frac{{a}_{0}}{{z}^{n}}}\right) \]\n\nwhenever \( z \neq... | Yes |
Corollary 4.7 Every polynomial \( P\left( z\right) = {a}_{n}{z}^{n} + \cdots + {a}_{0} \) of degree \( n \geq \) 1 has precisely \( n \) roots in \( \mathbb{C} \) . If these roots are denoted by \( {w}_{1},\ldots ,{w}_{n} \) , then \( P \) can be factored as \[ P\left( z\right) = {a}_{n}\left( {z - {w}_{1}}\right) \lef... | Proof. By the previous result \( P \) has a root, say \( {w}_{1} \) . Then, writing \( z = \left( {z - {w}_{1}}\right) + {w}_{1} \), inserting this expression for \( z \) in \( P \), and using the binomial formula we get \[ P\left( z\right) = {b}_{n}{\left( z - {w}_{1}\right) }^{n} + \cdots + {b}_{1}\left( {z - {w}_{1}... | Yes |
Theorem 4.8 Suppose \( f \) is a holomorphic function in a region \( \Omega \) that vanishes on a sequence of distinct points with a limit point in \( \Omega \) . Then \( f \) is identically 0 . | Proof. Suppose that \( {z}_{0} \in \Omega \) is a limit point for the sequence \( {\left\{ {w}_{k}\right\} }_{k = 1}^{\infty } \) and that \( f\left( {w}_{k}\right) = 0 \) . First, we show that \( f \) is identically zero in a small disc containing \( {z}_{0} \) . For that, we choose a disc \( D \) centered at \( {z}_{... | Yes |
Theorem 5.1 Suppose \( f \) is a continuous function in the open disc \( D \) such that for any triangle \( T \) contained in \( D \)\n\n\[{\int }_{T}f\left( z\right) {dz} = 0\]\n\nthen \( f \) is holomorphic. | Proof. By the proof of Theorem 2.1 the function \( f \) has a primitive \( F \) in \( D \) that satisfies \( {F}^{\prime } = f \) . By the regularity theorem, we know that \( F \) is indefinitely (and hence twice) complex differentiable, and therefore \( f \) is holomorphic. | Yes |
Theorem 5.2 If \( {\left\{ {f}_{n}\right\} }_{n = 1}^{\infty } \) is a sequence of holomorphic functions that converges uniformly to a function \( f \) in every compact subset of \( \Omega \), then \( f \) is holomorphic in \( \Omega \) . | Proof. Let \( D \) be any disc whose closure is contained in \( \Omega \) and \( T \) any triangle in that disc. Then, since each \( {f}_{n} \) is holomorphic, Goursat’s theorem implies\n\n\[{\int }_{T}{f}_{n}\left( z\right) {dz} = 0\;\text{ for all }n.\]\n\nBy assumption \( {f}_{n} \rightarrow f \) uniformly in the cl... | Yes |
Theorem 5.3 Under the hypotheses of the previous theorem, the sequence of derivatives \( {\left\{ {f}_{n}^{\prime }\right\} }_{n = 1}^{\infty } \) converges uniformly to \( {f}^{\prime } \) on every compact subset of \( \Omega \) . | Proof. We may assume without loss of generality that the sequence of functions in the theorem converges uniformly on all of \( \Omega \) . Given \( \delta > 0 \) , let \( {\Omega }_{\delta } \) denote the subset of \( \Omega \) defined by\n\n\[ \n{\Omega }_{\delta } = \left\{ {z \in \Omega : \overline{{D}_{\delta }}\le... | Yes |
Theorem 5.4 Let \( F\left( {z, s}\right) \) be defined for \( \left( {z, s}\right) \in \Omega \times \left\lbrack {0,1}\right\rbrack \) where \( \Omega \) is an open set in \( \mathbb{C} \) . Suppose \( F \) satisfies the following properties:\n\n(i) \( F\left( {z, s}\right) \) is holomorphic in \( z \) for each \( s \... | Proof. For each \( n \geq 1 \), we consider the Riemann sum\n\n\[ {f}_{n}\left( z\right) = \left( {1/n}\right) \mathop{\sum }\limits_{{k = 1}}^{n}F\left( {z, k/n}\right) .\n\nThen \( {f}_{n} \) is holomorphic in all of \( \Omega \) by property (i), and we claim that on any disc \( D \) whose closure is contained in \( ... | Yes |
Theorem 5.5 (Symmetry principle) If \( {f}^{ + } \) and \( {f}^{ - } \) are holomorphic functions in \( {\Omega }^{ + } \) and \( {\Omega }^{ - } \) respectively, that extend continuously to \( I \) and\n\n\[ \n{f}^{ + }\left( x\right) = {f}^{ - }\left( x\right) \;\text{ for all }x \in I, \n\]\n\nthen the function \( f... | Proof. One notes first that \( f \) is continuous throughout \( \Omega \) . The only difficulty is to prove that \( f \) is holomorphic at points of \( I \) . Suppose \( D \) is a disc centered at a point on \( I \) and entirely contained in \( \Omega \) . We prove that \( f \) is holomorphic in \( D \) by Morera’s the... | Yes |
Theorem 5.6 (Schwarz reflection principle) Suppose that \( f \) is a holomorphic function in \( {\Omega }^{ + } \) that extends continuously to \( I \) and such that \( f \) is real-valued on \( I \) . Then there exists a function \( F \) holomorphic in all of \( \Omega \) such that \( F = f \) on \( {\Omega }^{ + } \)... | Proof. The idea is simply to define \( F\left( z\right) \) for \( z \in {\Omega }^{ - } \) by\n\n\[ F\left( z\right) = \overline{f\left( \bar{z}\right) }.\]\n\nTo prove that \( F \) is holomorphic in \( {\Omega }^{ - } \) we note that if \( z,{z}_{0} \in {\Omega }^{ - } \), then \( \bar{z},\overline{{z}_{0}} \in {\Omeg... | Yes |
Lemma 5.8 Suppose \( f \) is holomorphic in an open set \( \Omega \), and \( K \subset \Omega \) is compact. Then, there exists finitely many segments \( {\gamma }_{1},\ldots ,{\gamma }_{N} \) in \( \Omega - K \) such that\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 1}}^{N}\frac{1}{2\pi i}{\int }_{{\gamma }_{n... | Proof. Let \( d = c \cdot d\left( {K,{\Omega }^{c}}\right) \), where \( c \) is any constant \( < 1/\sqrt{2} \), and consider a grid formed by (solid) squares with sides parallel to the axis and of length \( d \) .\n\nWe let \( \mathcal{Q} = \left\{ {{Q}_{1},\ldots ,{Q}_{M}}\right\} \) denote the finite collection of s... | Yes |
Lemma 5.9 For any line segment \( \gamma \) entirely contained in \( \Omega - K \), there exists a sequence of rational functions with singularities on \( \gamma \) that approximate the integral \( {\int }_{\gamma }f\left( \zeta \right) /\left( {\zeta - z}\right) {d\zeta } \) uniformly on \( K \) . | Proof. If \( \gamma \left( t\right) : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{C} \) is a parametrization for \( \gamma \), then\n\n\[ \n{\int }_{\gamma }\frac{f\left( \zeta \right) }{\zeta - z}{d\zeta } = {\int }_{0}^{1}\frac{f\left( {\gamma \left( t\right) }\right) }{\gamma \left( t\right) - z}{\gamma }^{\... | Yes |
Lemma 5.10 If \( {K}^{c} \) is connected and \( {z}_{0} \notin K \), then the function \( 1/\left( {z - {z}_{0}}\right) \) can be approximated uniformly on \( K \) by polynomials. | Proof. First, we choose a point \( {z}_{1} \) that is outside a large open disc \( D \) centered at the origin and which contains \( K \) . Then\n\n\[ \frac{1}{z - {z}_{1}} = - \frac{1}{{z}_{1}}\frac{1}{1 - z/{z}_{1}} = \mathop{\sum }\limits_{{n = 1}}^{\infty } - \frac{{z}^{n}}{{z}_{1}^{n + 1}} \]\n\nwhere the series c... | Yes |
Theorem 1.1 Suppose that \( f \) is holomorphic in a connected open set \( \Omega \) , has a zero at a point \( {z}_{0} \in \Omega \), and does not vanish identically in \( \Omega \) . Then there exists a neighborhood \( U \subset \Omega \) of \( {z}_{0} \), a non-vanishing holomorphic function \( g \) on \( U \), and ... | Proof. Since \( \Omega \) is connected and \( f \) is not identically zero, we conclude that \( f \) is not identically zero in a neighborhood of \( {z}_{0} \) . In a small disc centered at \( {z}_{0} \) the function \( f \) has a power series expansion\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }... | Yes |
Theorem 1.2 If \( f \) has a pole at \( {z}_{0} \in \Omega \), then in a neighborhood of that point there exist a non-vanishing holomorphic function \( h \) and a unique positive integer \( n \) such that\n\n\[ f\left( z\right) = {\left( z - {z}_{0}\right) }^{-n}h\left( z\right) . \] | Proof. By the previous theorem we have \( 1/f\left( z\right) = {\left( z - {z}_{0}\right) }^{n}g\left( z\right) \) , where \( g \) is holomorphic and non-vanishing in a neighborhood of \( {z}_{0} \), so the result follows with \( h\left( z\right) = 1/g\left( z\right) \) . | Yes |
Theorem 1.3 If \( f \) has a pole of order \( n \) at \( {z}_{0} \), then\n\n\[ f\left( z\right) = \frac{{a}_{-n}}{{\left( z - {z}_{0}\right) }^{n}} + \frac{{a}_{-n + 1}}{{\left( z - {z}_{0}\right) }^{n - 1}} + \cdots + \frac{{a}_{-1}}{\left( z - {z}_{0}\right) } + G\left( z\right) ,\] | Proof. The proof follows from the multiplicative statement in the previous theorem. Indeed, the function \( h \) has a power series expansion\n\n\[ h\left( z\right) = {A}_{0} + {A}_{1}\left( {z - {z}_{0}}\right) + \cdots \]\n\nso that\n\n\[ f\left( z\right) = {\left( z - {z}_{0}\right) }^{-n}\left( {{A}_{0} + {A}_{1}\l... | Yes |
Theorem 1.4 If \( f \) has a pole of order \( n \) at \( {z}_{0} \), then\n\n\[ \n{\operatorname{res}}_{{z}_{0}}f = \mathop{\lim }\limits_{{z \rightarrow {z}_{0}}}\frac{1}{\left( {n - 1}\right) !}{\left( \frac{d}{dz}\right) }^{n - 1}{\left( z - {z}_{0}\right) }^{n}f\left( z\right) .\n\] | The theorem is an immediate consequence of formula (1), which implies\n\n\[ \n{\left( z - {z}_{0}\right) }^{n}f\left( z\right) = {a}_{-n} + {a}_{-n + 1}\left( {z - {z}_{0}\right) + \cdots + {a}_{-1}{\left( z - {z}_{0}\right) }^{n - 1} + \n\]\n\n\[ \n+ G\left( z\right) {\left( z - {z}_{0}\right) }^{n}.\n\] | Yes |
Theorem 2.1 Suppose that \( f \) is holomorphic in an open set containing a circle \( C \) and its interior, except for a pole at \( {z}_{0} \) inside \( C \) . Then\n\n\[ \n{\int }_{C}f\left( z\right) {dz} = {2\pi i}{\operatorname{res}}_{{z}_{0}}f. \n\] | Proof. Once again, we may choose a keyhole contour that avoids the pole, and let the width of the corridor go to zero to see that\n\n\[ \n{\int }_{C}f\left( z\right) {dz} = {\int }_{{C}_{\epsilon }}f\left( z\right) {dz} \n\]\n\nwhere \( {C}_{\epsilon } \) is the small circle centered at the pole \( {z}_{0} \) and of ra... | Yes |
Corollary 2.2 Suppose that \( f \) is holomorphic in an open set containing a circle \( C \) and its interior, except for poles at the points \( {z}_{1},\ldots ,{z}_{N} \) inside C. Then\n\n\[{\int }_{C}f\left( z\right) {dz} = {2\pi i}\mathop{\sum }\limits_{{k = 1}}^{N}{\operatorname{res}}_{{z}_{k}}f.\] | For the proof, consider a multiple keyhole which has a loop avoiding each one of the poles. Let the width of the corridors go to zero. In the limit, the integral over the large circle equals a sum of integrals over small circles to which Theorem 2.1 applies. | No |
Corollary 2.3 Suppose that \( f \) is holomorphic in an open set containing a toy contour \( \gamma \) and its interior, except for poles at the points \( {z}_{1},\ldots ,{z}_{N} \) inside \( \gamma \) . Then\n\n\[ \n{\int }_{\gamma }f\left( z\right) {dz} = {2\pi i}\mathop{\sum }\limits_{{k = 1}}^{N}{\operatorname{res}... | The proof consists of choosing a keyhole appropriate for the given toy contour, so that, as we have seen previously, we can reduce the situation to integrating over small circles around the poles where Theorem 2.1 applies. | No |
Theorem 3.1 (Riemann's theorem on removable singularities) Suppose that \( f \) is holomorphic in an open set \( \Omega \) except possibly at a point \( {z}_{0} \) in \( \Omega \) . If \( f \) is bounded on \( \Omega - \left\{ {z}_{0}\right\} \), then \( {z}_{0} \) is a removable singularity. | Proof. Since the problem is local we may consider a small disc \( D \) centered at \( {z}_{0} \) and whose closure is contained in \( \Omega \) . Let \( C \) denote the boundary circle of that disc with the usual positive orientation. We shall prove that if \( z \in D \) and \( z \neq {z}_{0} \), then under the assumpt... | Yes |
Corollary 3.2 Suppose that \( f \) has an isolated singularity at the point \( {z}_{0} \) . Then \( {z}_{0} \) is a pole of \( f \) if and only if \( \left| {f\left( z\right) }\right| \rightarrow \infty \) as \( z \rightarrow {z}_{0} \) . | Proof. If \( {z}_{0} \) is a pole, then we know that \( 1/f \) has a zero at \( {z}_{0} \), and therefore \( \left| {f\left( z\right) }\right| \rightarrow \infty \) as \( z \rightarrow {z}_{0} \) . Conversely, suppose that this condition holds. Then \( 1/f \) is bounded near \( {z}_{0} \), and in fact \( 1/\left| {f\le... | Yes |
Theorem 3.3 (Casorati-Weierstrass) Suppose \( f \) is holomorphic in the punctured disc \( {D}_{r}\left( {z}_{0}\right) - \left\{ {z}_{0}\right\} \) and has an essential singularity at \( {z}_{0} \) . Then, the image of \( {D}_{r}\left( {z}_{0}\right) - \left\{ {z}_{0}\right\} \) under \( f \) is dense in the complex p... | Proof. We argue by contradiction. Assume that the range of \( f \) is not dense, so that there exists \( w \in \mathbb{C} \) and \( \delta > 0 \) such that\n\n\[ \left| {f\left( z\right) - w}\right| > \delta \;\text{ for all }z \in {D}_{r}\left( {z}_{0}\right) - \left\{ {z}_{0}\right\} . \]\n\nWe may therefore define a... | Yes |
Theorem 3.4 The meromorphic functions in the extended complex plane are the rational functions. | Proof. Suppose that \( f \) is meromorphic in the extended plane. Then \( f\left( {1/z}\right) \) has either a pole or a removable singularity at 0, and in either case it must be holomorphic in a deleted neighborhood of the origin, so that the function \( f \) can have only finitely many poles in the plane, say at \( {... | Yes |
Theorem 4.3 (Rouché’s theorem) Suppose that \( f \) and \( g \) are holomorphic in an open set containing a circle \( C \) and its interior. If\n\n\[ \left| {f\left( z\right) }\right| > \left| {g\left( z\right) }\right| \;\text{ for all }z \in C, \]\n\nthen \( f \) and \( f + g \) have the same number of zeros inside t... | Proof. For \( t \in \left\lbrack {0,1}\right\rbrack \) define\n\n\[ {f}_{t}\left( z\right) = f\left( z\right) + \operatorname{tg}\left( z\right) \]\n\nso that \( {f}_{0} = f \) and \( {f}_{1} = f + g \) . Let \( {n}_{t} \) denote the number of zeros of \( {f}_{t} \) inside the circle counted with multiplicities, so tha... | Yes |
Theorem 4.4 (Open mapping theorem) If \( f \) is holomorphic and nonconstant in a region \( \Omega \), then \( f \) is open. | Proof. Let \( {w}_{0} \) belong to the image of \( f \), say \( {w}_{0} = f\left( {z}_{0}\right) \) . We must prove that all points \( w \) near \( {w}_{0} \) also belong to the image of \( f \) . Define \( g\left( z\right) = f\left( z\right) - w \) and write \[ g\left( z\right) = \left( {f\left( z\right) - {w}_{0}}\ri... | Yes |
Theorem 4.5 (Maximum modulus principle) If \( f \) is a non-constant holomorphic function in a region \( \Omega \), then \( f \) cannot attain a maximum in \( \Omega \) . | Proof. Suppose that \( f \) did attain a maximum at \( {z}_{0} \) . Since \( f \) is holomorphic it is an open mapping, and therefore, if \( D \subset \Omega \) is a small disc centered at \( {z}_{0} \), its image \( f\left( D\right) \) is open and contains \( f\left( {z}_{0}\right) \) . This proves that there are poin... | Yes |
Corollary 4.6 Suppose that \( \Omega \) is a region with compact closure \( \bar{\Omega } \) . If \( f \) is holomorphic on \( \Omega \) and continuous on \( \bar{\Omega } \) then\n\n\[ \mathop{\sup }\limits_{{z \in \Omega }}\left| {f\left( z\right) }\right| \leq \mathop{\sup }\limits_{{z \in \bar{\Omega } - \Omega }}\... | In fact, since \( f\left( z\right) \) is continuous on the compact set \( \bar{\Omega } \), then \( \left| {f\left( z\right) }\right| \) attains its maximum in \( \bar{\Omega } \) ; but this cannot be in \( \Omega \) if \( f \) is non-constant. If \( f \) is constant, the conclusion is trivial. | Yes |
Theorem 5.2 Any holomorphic function in a simply connected domain has a primitive. | Proof. Fix a point \( {z}_{0} \) in \( \Omega \) and define\n\n\[ F\left( z\right) = {\int }_{\gamma }f\left( w\right) {dw} \]\n\nwhere the integral is taken over any curve in \( \Omega \) joining \( {z}_{0} \) to \( z \) . This definition is independent of the curve chosen, since \( \Omega \) is simply connected, and ... | Yes |
Corollary 5.3 If \( f \) is holomorphic in the simply connected region \( \Omega \) , then\n\n\[{\int }_{\gamma }f\left( z\right) {dz} = 0\]\n\nfor any closed curve \( \gamma \) in \( \Omega \) . | This is immediate from the existence of a primitive. | No |
Theorem 6.1 Suppose that \( \Omega \) is simply connected with \( 1 \in \Omega \), and \( 0 \notin \) \( \Omega \) . Then in \( \Omega \) there is a branch of the logarithm \( F\left( z\right) = {\log }_{\Omega }\left( z\right) \) so that\n\n(i) \( F \) is holomorphic in \( \Omega \) ,\n\n(ii) \( {e}^{F\left( z\right) ... | Proof. We shall construct \( F \) as a primitive of the function \( 1/z \) . Since \( 0 \notin \Omega \), the function \( f\left( z\right) = 1/z \) is holomorphic in \( \Omega \) . We define\n\n\[ \n{\log }_{\Omega }\left( z\right) = F\left( z\right) = {\int }_{\gamma }f\left( w\right) {dw} \n\]\n\nwhere \( \gamma \) i... | Yes |
Theorem 6.2 If \( f \) is a nowhere vanishing holomorphic function in a simply connected region \( \Omega \), then there exists a holomorphic function \( g \) on \( \Omega \) such that\n\n\[ f\left( z\right) = {e}^{g\left( z\right) }.\] | Proof. Fix a point \( {z}_{0} \) in \( \Omega \), and define a function\n\n\[ g\left( z\right) = {\int }_{\gamma }\frac{{f}^{\prime }\left( w\right) }{f\left( w\right) }{dw} + {c}_{0} \]\n\nwhere \( \gamma \) is any path in \( \Omega \) connecting \( {z}_{0} \) to \( z \), and \( {c}_{0} \) is a complex number so that ... | Yes |
Theorem 7.1 The coefficients of the power series expansion of \( f \) are given by\n\n\[ \n{a}_{n} = \frac{1}{{2\pi }{r}^{n}}{\int }_{0}^{2\pi }f\left( {{z}_{0} + r{e}^{i\theta }}\right) {e}^{-{in\theta }}{d\theta }\n\]\n\nfor all \( n \geq 0 \) and \( 0 < r < R \) . Moreover,\n\n\[ \n0 = \frac{1}{{2\pi }{r}^{n}}{\int ... | Proof. Since \( {f}^{\left( n\right) }\left( {z}_{0}\right) = {a}_{n}n \) !, the Cauchy integral formula gives\n\n\[ \n{a}_{n} = \frac{1}{2\pi i}{\int }_{\gamma }\frac{f\left( \zeta \right) }{{\left( \zeta - {z}_{0}\right) }^{n + 1}}{d\zeta }\n\]\n\nwhere \( \gamma \) is a circle of radius \( 0 < r < R \) centered at \... | Yes |
Corollary 7.3 If \( f \) is holomorphic in a disc \( {D}_{R}\left( {z}_{0}\right) \), and \( u = \operatorname{Re}\left( f\right) \) , then\n\n\[ u\left( {z}_{0}\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }u\left( {{z}_{0} + r{e}^{i\theta }}\right) {d\theta },\;\text{ for any }0 < r < R. \] | Recall that \( u \) is harmonic whenever \( f \) is holomorphic, and in fact, the above corollary is a property enjoyed by every harmonic function in the disc \( {D}_{R}\left( {z}_{0}\right) \) . This follows from Exercise 12 in Chapter 2, which shows that every harmonic function in a disc is the real part of a holomor... | No |
Theorem 2.1 If \( f \) belongs to the class \( {\mathfrak{F}}_{a} \) for some \( a > 0 \), then \( \left| {\widehat{f}\left( \xi \right) }\right| \leq B{e}^{-{2\pi b}\left| \xi \right| } \) for any \( 0 \leq b < a \) . | Proof. Recall that \( \widehat{f}\left( \xi \right) = {\int }_{-\infty }^{\infty }f\left( x\right) {e}^{-{2\pi ix\xi }}{dx} \). The case \( b = 0 \) simply says that \( \widehat{f} \) is bounded, which follows at once from the integral defining \( \widehat{f} \), the assumption that \( f \) is of moderate decrease, and... | Yes |
Lemma 2.3 If \( A \) is positive and \( B \) is real, then \( {\int }_{0}^{\infty }{e}^{-\left( {A + {iB}}\right) \xi }{d\xi } = \) \( \frac{1}{A + {iB}} \) . | Proof. Since \( A > 0 \) and \( B \in \mathbb{R} \), we have \( \left| {e}^{-\left( {A + {iB}}\right) \xi }\right| = {e}^{-{A\xi }} \), and the integral converges. By definition\n\n\[ \n{\int }_{0}^{\infty }{e}^{-\left( {A + {iB}}\right) \xi }{d\xi } = \mathop{\lim }\limits_{{R \rightarrow \infty }}{\int }_{0}^{R}{e}^{... | Yes |
Theorem 3.1 Suppose \( \widehat{f} \) satisfies the decay condition \( \left| {\widehat{f}\left( \xi \right) }\right| \leq A{e}^{-{2\pi a}\left| \xi \right| } \) for some constants \( a, A > 0 \) . Then \( f\left( x\right) \) is the restriction to \( \mathbb{R} \) of \( a \) function \( f\left( z\right) \) holomorphic ... | Proof. Define\n\n\[ \n{f}_{n}\left( z\right) = {\int }_{-n}^{n}\widehat{f}\left( \xi \right) {e}^{2\pi i\xi z}{d\xi }\n\]\n\nand note that \( {f}_{n} \) is entire by Theorem 5.4 in Chapter 2. Observe also that \( f\left( z\right) \) may be defined for all \( z \) in the strip \( {S}_{b} \) by\n\n\[ \nf\left( z\right) =... | Yes |
Corollary 3.2 If \( \widehat{f}\left( \xi \right) = O\left( {e}^{-{2\pi a}\left| \xi \right| }\right) \) for some \( a > 0 \), and \( f \) vanishes in a non-empty open interval, then \( f = 0 \) . | Since by the theorem \( f \) is analytic in a region containing the real line, the corollary is a consequence of Theorem 4.8 in Chapter 2. In particular, we recover the fact proved in Exercise 21, Chapter 5 in Book I, namely that \( f \) and \( \widehat{f} \) cannot both have compact support unless \( f = 0 \) . | No |
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