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Consider the function \( f : {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) given by \( f\left( {x, y}\right) = {2x} + {y}^{2} \). | \[ f\left( {x + {\lambda u}, y + {\lambda v}}\right) = 2\left( {x + {\lambda u}}\right) + {\left( y + \lambda v\right) }^{2} = {2x} + {y}^{2} + 2\left( {u + {yv}}\right) \lambda + {v}^{2}{\lambda }^{2}, \] if \( v \neq 0 \), we see that the above quadratic function of \( \lambda \) increases for \( \lambda \geq - \left... | Yes |
Proposition 51.36. Let \( f \) be a proper and closed convex function over \( {\mathbb{R}}^{n} \). The function \( h \) given by \( h\left( x\right) = f\left( x\right) + q\left( x\right) \) obtained by adding any strictly convex quadratic function \( q \) of the form \( q\left( x\right) = {x}^{\top }{Ax} + {b}^{\top }x... | Proof. By Theorem 51.13 there is some affine form \( \varphi \) given by \( \varphi \left( x\right) = {c}^{\top }x + \alpha \) (with \( \alpha \in \mathbb{R} \) ) such that \( f\left( x\right) \geq \varphi \left( x\right) \) for all \( x \in {\mathbb{R}}^{n} \). Then we have\n\n\[ h\left( x\right) = f\left( x\right) + ... | Yes |
Proposition 51.37. Let \( h \) be a proper convex function on \( {\mathbb{R}}^{n} \), and let \( C \) be a nonempty convex subset of \( {\mathbb{R}}^{n} \). (1) For any \( x \in {\mathbb{R}}^{n} \), if there is some \( y \in \partial h\left( x\right) \) such that \( - y \in {N}_{C}\left( x\right) \), that is, \( - y \)... | Proof. (1) By Proposition 51.33, \( h \) attains its minimum on \( C \) at \( x \) iff \[ 0 \in \partial \left( {h + {I}_{C}}\right) \left( x\right) \] By Proposition 51.22, since \[ \partial \left( {h + {I}_{C}}\right) \left( x\right) \subseteq \partial h\left( x\right) + \partial {I}_{C}\left( x\right) \] if \( 0 \in... | Yes |
Theorem 51.40. (Theorem 28.3, Rockafellar) Let (P) be an ordinary convex program. If \( x \in {\mathbb{R}}^{n} \) and \( \left( {\lambda ,\mu }\right) \in {\mathbb{R}}_{ + }^{m} \times {\mathbb{R}}^{p} \), then \( \left( {\lambda ,\mu }\right) \) and \( x \) have the property that\n\n(a) The infimum of the function \( ... | Moreover, this condition holds iff the following KKT conditions hold:\n\n(1) \( \lambda \in {\mathbb{R}}_{ + }^{m},{\varphi }_{i}\left( x\right) \leq 0 \), and \( {\lambda }_{i}{\varphi }_{i}\left( x\right) = 0 \) for \( i = 1,\ldots, m \) .\n\n(2) \( {\psi }_{j}\left( x\right) = 0 \) for \( j = 1,\ldots p \) .\n\n(3) ... | Yes |
Theorem 51.42. (Theorem 28.4, Rockafellar) Let (P) be an ordinary convex program with Lagrangian \( L\left( {x,\lambda ,\mu }\right) \) . If the Lagrange multipliers \( \left( {{\lambda }^{ * },{\mu }^{ * }}\right) \in {\mathbb{R}}_{ + }^{m} \times {\mathbb{R}}^{p} \) and the vector \( {x}^{ * } \in {\mathbb{R}}^{n} \)... | More generally, the Lagrange multipliers \( \left( {{\lambda }^{ * },{\mu }^{ * }}\right) \in {\mathbb{R}}_{ + }^{m} \times {\mathbb{R}}^{p} \) have Property (a) iff\n\n\[ \n- \infty < \mathop{\inf }\limits_{x}L\left( {x,{\lambda }^{ * },{\mu }^{ * }}\right) \leq \mathop{\sup }\limits_{{\lambda ,\mu }}\mathop{\inf }\li... | Yes |
Let us look at a very simple example of the gradient ascent method applied to a problem we first encountered in Section 42.1, namely minimize \( J\left( {x, y}\right) = \left( {1/2}\right) \left( {{x}^{2} + {y}^{2}}\right) \) subject to \( {2x} - y = 5 \) . | The Lagrangian is\n\n\[ L\left( {x, y,\lambda }\right) = \frac{1}{2}\left( {{x}^{2} + {y}^{2}}\right) + \lambda \left( {{2x} - y - 5}\right) . \]\n\nThe method of Lagrangian duality says first calculate\n\n\[ G\left( \lambda \right) = \mathop{\inf }\limits_{{\left( {x, y}\right) \in {\mathbb{R}}^{2}}}L\left( {x, y,\lam... | Yes |
Example 52.4. Let us reconsider the problem of Example 52.2 to solve it using ADMM. We formulate the problem as\n\n\[ \n\\text{minimize}{2x} + {z}^{2} \n\]\n\n\[ \n\\text{subject to}{2x} - z = 0\\text{,}\n\]\n\nwith \( f\\left( x\\right) = {2x} \) and \( g\\left( z\\right) = {z}^{2} \) . The augmented Lagrangian is giv... | The ADMM steps are as follows. The \( x \) -update is\n\n\[ \n{x}^{k + 1} = \\underset{x}{\\arg \\min }\\left( {{2\\rho }{x}^{2} - {2\\rho x}{z}^{k} + 2{\\lambda }^{k}x + {2x}}\\right) ,\n\]\n\nand since this is a quadratic function in \( x \), its minimum is achieved when the derivative of the above function (with res... | Yes |
Theorem 52.1. Suppose the following assumptions hold:\n\n(1) The functions \( f : \mathbb{R} \rightarrow \mathbb{R} \cup \{ + \infty \} \) and \( g : \mathbb{R} \rightarrow \mathbb{R} \cup \{ + \infty \} \) are proper and closed convex functions (see Section 51.1) such that \( \operatorname{relint}\left( {\operatorname... | Proof. The core of the proof is due to Boyd et al. [28], but there are new steps because we have the stronger hypothesis (2), which yield the stronger result (3).\n\nThe proof consists of several steps. It is not possible to prove directly that the sequences \( \left( {x}^{k}\right) ,\left( {z}^{k}\right) \), and \( \l... | Yes |
Example 52.5. Let\n\n\[ f\left( {x, y}\right) = x,\;g\left( z\right) = 0,\;y - z = 0. \]\n\nThen\n\n\[ {L}_{\rho }\left( {x, y, z,\lambda }\right) = x + \lambda \left( {y - z}\right) + \left( {\rho /2}\right) {\left( y - z\right) }^{2}, \] | but minimizing over \( \left( {x, y}\right) \) with \( z \) held constant yields \( - \infty \), which implies that the above program has no finite optimal solution. See Figure 52.4. | Yes |
Consider the problem given by\n\n\[ f\left( x\right) = x,\;g\left( z\right) = 0,\;x - z = 0. \] | Since \( f\left( x\right) + g\left( z\right) = x \), and \( x = z \), the variable \( x \) is unconstrained and the above function goes to \( - \infty \) when \( x \) goes to \( - \infty \) . The augmented Lagrangian is\n\n\[ {L}_{\rho }\left( {x, z,\lambda }\right) = x + \lambda \left( {x - z}\right) + \frac{\rho }{2}... | Yes |
A first simplification arises when \( A = I \), in which case the \( x \) -update is\n\n\[ \n{x}^{ + } = \underset{x}{\arg \min }\left( {f\left( x\right) + \left( {\rho /2}\right) \parallel x - v{\parallel }_{2}^{2}}\right) = {\operatorname{prox}}_{f,\rho }\left( v\right) .\n\] | The map \( v \mapsto {\operatorname{prox}}_{f,\rho }\left( v\right) \) is known as the proximity operator of \( f \) with penalty \( \rho \) . The above minimization is generally referred to as proximal minimization. | Yes |
When the function \( f \) is simple enough, the proximity operator can be computed analytically. This is the case in particular when \( f = {I}_{C} \), the indicator function of a nonempty closed convex set \( C \). | In this case, it is easy to see that\n\n\[ \n{x}^{ + } = \underset{x}{\arg \min }\left( {{I}_{C}\left( x\right) + \left( {\rho /2}\right) \parallel x - v{\parallel }_{2}^{2}}\right) = {\Pi }_{C}\left( v\right) , \n\]\n\nthe orthogonal projection of \( v \) onto \( C \). In the special case where \( C = {\mathbb{R}}_{ +... | Yes |
A second case where simplifications arise is the case where \( f \) is a convex quadratic functional of the form\n\n\[ f\left( x\right) = \frac{1}{2}{x}^{\top }{Px} + {q}^{\top }x + r \]\n\nwhere \( P \) is a \( n \times n \) symmetric positive semidefinite matrix, \( q \in {\mathbb{R}}^{n} \) and \( r \in \mathbb{R} \... | and since \( A \) has rank \( n \), the matrix \( {A}^{\top }A \) is symmetric positive definite, so we get\n\n\[ {x}^{ + } = {\left. \left( P + \rho {A}^{\top }A\right) \right) }^{-1}\left( {\rho {A}^{\top }v - q}\right) . \] | Yes |
A third case where simplifications arise is the variation of the previous case where \( f \) is a convex quadratic functional of the form\n\n\[ f\left( x\right) = \frac{1}{2}{x}^{\top }{Px} + {q}^{\top }x + r, \]\n\nexcept that \( f \) is constrained by equality constraints \( {Cx} = b \), as in Section 50.4, which mea... | so by the results of Section 50.4, \( {x}^{ + } \) is a component of the solution of the KKT-system\n\n\[ \left( \begin{matrix} P + {\rho I} & {C}^{\top } \\ C & 0 \end{matrix}\right) \left( \begin{matrix} {x}^{ + } \\ \lambda \end{matrix}\right) = \left( \begin{matrix} - q + {\rho v} \\ b \end{matrix}\right) . \]\n\nT... | Yes |
Suppose we have two feature maps \( {\varphi }_{1} : X \rightarrow {\mathbb{R}}^{{n}_{1}} \) and \( {\varphi }_{2} : X \rightarrow {\mathbb{R}}^{{n}_{2}} \), and let \( {\kappa }_{1}\left( {x, y}\right) = \left\langle {{\varphi }_{1}\left( x\right) ,{\varphi }_{1}\left( y\right) }\right\rangle \) and \( {\kappa }_{2}\l... | \[ \langle \varphi \left( x\right) ,\varphi \left( y\right) \rangle = \left\langle {\left( {{\varphi }_{1}\left( x\right) ,{\varphi }_{2}\left( x\right) }\right) ,\left( {{\varphi }_{1}\left( y\right) ,{\varphi }_{2}\left( y\right) }\right) }\right\rangle = \left\langle {{\varphi }_{1}\left( x\right) ,{\varphi }_{1}\le... | Yes |
Let \( X \) be a subset of \( {\mathbb{R}}^{2} \), and let \( {\varphi }_{1} : X \rightarrow {\mathbb{R}}^{3} \) be the map given by\n\n\[ \n{\varphi }_{1}\left( {{x}_{1},{x}_{2}}\right) = \left( {{x}_{1}^{2},{x}_{2}^{2},\sqrt{2}{x}_{1}{x}_{2}}\right) .\n\] | Observe that linear relations in the feature space \( H = {\mathbb{R}}^{3} \) correspond to quadratic relations in the input space (of data). We have\n\n\[ \n\left\langle {{\varphi }_{1}\left( x\right) ,{\varphi }_{1}\left( y\right) }\right\rangle = \left\langle {\left( {{x}_{1}^{2},{x}_{2}^{2},\sqrt{2}{x}_{1}{x}_{2}}\... | Yes |
Example 54.3. Example 54.2 can be generalized as follows. Suppose we have a feature map \( {\varphi }_{1} : X \rightarrow {\mathbb{R}}^{n} \) and let \( {\kappa }_{1}\left( {x, y}\right) = \left\langle {{\varphi }_{1}\left( x\right) ,{\varphi }_{1}\left( y\right) }\right\rangle \) be the corresponding kernel function (... | \[ \n= \mathop{\sum }\limits_{{i, j = 1}}^{n}{\left( {\varphi }_{1}\left( x\right) \right) }_{i}{\left( {\varphi }_{1}\left( x\right) \right) }_{j}{\left( {\varphi }_{1}\left( y\right) \right) }_{i}{\left( {\varphi }_{1}\left( y\right) \right) }_{j} \n\]\n\n\[ \n= \mathop{\sum }\limits_{{i = 1}}^{n}{\left( {\varphi }_{... | Yes |
Note that the feature map \( \varphi : X \rightarrow {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \) is not very economical because if \( i \neq j \) then the components \( {\varphi }_{\left( i, j\right) }\left( x\right) \) and \( {\varphi }_{\left( j, i\right) }\left( x\right) \) are both equal to \( {\left( {\varphi }_{1... | \[ {\varphi }^{\prime }{\left( x\right) }_{\left( i, j\right) } = \left\{ \begin{array}{ll} {\left( {\varphi }_{1}\left( x\right) \right) }_{i}^{2} & i = j, \\ \sqrt{2}{\left( {\varphi }_{1}\left( x\right) \right) }_{i}{\left( {\varphi }_{1}\left( x\right) \right) }_{j} & i < j, \end{array}\right. \] where the pairs \(... | Yes |
For any positive real constant \( R > 0 \), the constant function \( \kappa \left( {x, y}\right) = R \) is a kernel function corresponding to the feature map \( \varphi : X \rightarrow \mathbb{R} \) given by \( \varphi \left( {x, y}\right) = \sqrt{R} \) . | By definition, the function \( {\kappa }_{1}^{\prime } : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) given by \( {\kappa }_{1}^{\prime }\left( {x, y}\right) = \langle x, y\rangle \) is a kernel function (the feature map is the identity map from \( {\mathbb{R}}^{n} \) to itself). We just saw that for any positive real co... | Yes |
For any two subsets \( {A}_{1} \) and \( {A}_{2} \) of \( D \), it is easy to check that \[ \left\langle {\varphi \left( {A}_{1}\right) ,\varphi \left( {A}_{2}\right) }\right\rangle = {2}^{\left| {A}_{1} \cap {A}_{2}\right| } \] the number of common subsets of \( {A}_{1} \) and \( {A}_{2} \). | For example, \( {A}_{1} \cap {A}_{2} = \{ 2,3\} \), and \[ \left\langle {\varphi \left( {A}_{1}\right) ,\varphi \left( {A}_{2}\right) }\right\rangle = 4 \] Therefore, the function \( \kappa : X \times X \rightarrow \mathbb{R} \) given by \[ \kappa \left( {{A}_{1},{A}_{2}}\right) = {2}^{\left| {A}_{1} \cap {A}_{2}\right... | No |
Proposition 54.1. Let \( X \) be any nonempty set, let \( H \) be any (complex) Hilbert space, let \( \varphi : X \rightarrow H \) be any function, and let \( \kappa : X \times X \rightarrow \mathbb{C} \) be the kernel given by \[ \kappa \left( {x, y}\right) = \langle \varphi \left( x\right) ,\varphi \left( y\right) \r... | Proof. We have \[ {u}^{ * }{K}_{S}u = {u}^{\top }{K}_{S}^{\top }\bar{u} = \mathop{\sum }\limits_{{i, j = 1}}^{p}\kappa \left( {{x}_{i},{x}_{j}}\right) {u}_{i}\overline{{u}_{j}} \] \[ = \mathop{\sum }\limits_{{i, j = 1}}^{p}\langle \varphi \left( x\right) ,\varphi \left( y\right) \rangle {u}_{i}\overline{{u}_{j}} \] \[ ... | Yes |
Proposition 54.2. Let \( \kappa : X \times X \rightarrow \mathbb{C} \) be a positive definite kernel. Then \( \kappa \left( {x, x}\right) \geq 0 \) for all \( x \in X \), and for any finite subset \( S = \left\{ {{x}_{1},\ldots ,{x}_{p}}\right\} \) of \( X \), the \( p \times p \) matrix \( {K}_{S} \) given by\n\n\[ \n... | Proof. The first property is obvious by choosing \( S = \{ x\} \) . We have\n\n\[ \n{\left( u + v\right) }^{ * }{K}_{S}\left( {u + v}\right) = {u}^{ * }{K}_{S}u + {u}^{ * }{K}_{S}v + {v}^{ * }{K}_{S}u + {v}^{ * }{K}_{S}v,\n\]\n\nand since \( {\left( u + v\right) }^{ * }{K}_{S}\left( {u + v}\right) ,{u}^{ * }{K}_{S}u,{v... | Yes |
Proposition 54.3. If \( \kappa : X \times X \rightarrow \mathbb{R} \), then \( \kappa \) is a positive definite kernel iff for any finite subset \( S = \left\{ {{x}_{1},\ldots ,{x}_{p}}\right\} \) of \( X \), the \( p \times p \) real matrix \( {K}_{S} \) given by\n\n\[ \n{K}_{S} = {\left( \kappa \left( {x}_{k},{x}_{j}... | Proof. If \( \kappa \) is a real-valued positive definite kernel, then the proposition is a trivial consequence of Proposition 54.2.\n\nFor the converse, assume that \( \kappa \) is symmetric and that it satisfies the second condition of the proposition. We need to show that \( \kappa \) is a positive definite kernel w... | Yes |
Proposition 54.4. A hermitian \( 2 \times 2 \) matrix\n\n\[ A = \left( \begin{array}{ll} a & \bar{b} \\ b & d \end{array}\right) \]\n\nis positive semidefinite if and only if \( a \geq 0, d \geq 0 \), and ad \( - {\left| b\right| }^{2} \geq 0 \) . | Proof. For all \( x, y \in \mathbb{C} \), we have\n\n\[ \left( \begin{array}{ll} \overline{x} & \overline{y} \end{array}\right) \left( \begin{array}{ll} a & \overline{b} \\ b & d \end{array}\right) \left( \begin{array}{l} x \\ y \end{array}\right) = \left( \begin{array}{ll} \overline{x} & \overline{y} \end{array}\right... | Yes |
Proposition 54.5. (I. Schur) If \( {\kappa }_{1} : X \times X \rightarrow \mathbb{C} \) and \( {\kappa }_{2} : X \times X \rightarrow \mathbb{C} \) are two positive definite kernels, then the function \( \kappa : X \times X \rightarrow \mathbb{C} \) given by \( \kappa \left( {x, y}\right) = {\kappa }_{1}\left( {x, y}\r... | Proof. It suffices to prove that if \( A = \left( {a}_{jk}\right) \) and \( B = \left( {b}_{jk}\right) \) are two hermitian positive semidefinite \( p \times p \) matrices, then so is their pointwise product \( C = A \circ B = \left( {{a}_{jk}{b}_{jk}}\right) \) (also known as Hadamard or Schur product). Recall that a ... | Yes |
Proposition 54.6. Let \( {\kappa }_{1} : X \times X \rightarrow \mathbb{C} \) and \( {\kappa }_{2} : X \times X \rightarrow \mathbb{C} \) be two positive definite kernels, \( f : X \rightarrow \mathbb{C} \) be a function, \( \psi : X \rightarrow {\mathbb{R}}^{N} \) be a function, \( {\kappa }_{3} : {\mathbb{R}}^{N} \ti... | Proof. (1) For every finite subset \( S = \left\{ {{x}_{1},\ldots ,{x}_{p}}\right\} \) of \( X \), if \( {K}_{1} \) is the \( p \times p \) matrix\n\n\[ {K}_{1} = {\left( {\kappa }_{1}\left( {x}_{k},{x}_{j}\right) \right) }_{1 \leq j, k \leq p} \]\n\nand if \( {K}_{2} \) is the \( p \times p \) matrix\n\n\[ {K}_{2} = {... | Yes |
Proposition 54.7. Let \( {\kappa }_{1} : X \times X \rightarrow \mathbb{C} \) be a positive definite kernel, and let \( p\left( z\right) \) be a polynomial with nonnegative coefficients. Then the following functions \( \kappa \) defined below are also positive definite kernels.\n\n(1) \( \kappa \left( {x, y}\right) = p... | Proof. (1) If \( p\left( z\right) = {a}_{m}{z}^{m} + \cdots + {a}_{1}z + {a}_{0} \), then\n\n\[ p\left( {{\kappa }_{1}\left( {x, y}\right) }\right) = {a}_{m}{\kappa }_{1}{\left( x, y\right) }^{m} + \cdots + {a}_{1}{\kappa }_{1}\left( {x, y}\right) + {a}_{0}. \]\n\nSince \( {a}_{k} \geq 0 \) for \( k = 0,\ldots, m \), b... | Yes |
Theorem 54.8. Let \( \kappa : X \times X \rightarrow \mathbb{C} \) be a positive definite kernel on a nonempty set \( X \) . For every \( x \in X \), let \( {\kappa }_{x} : X \rightarrow \mathbb{C} \) be the function given by\n\n\[ \n{\kappa }_{x}\left( y\right) = \kappa \left( {x, y}\right) ,\;y \in X.\n\]\n\nLet \( {... | Proof. For any two linear combinations \( f = \mathop{\sum }\limits_{{j = 1}}^{p}{\alpha }_{j}{\kappa }_{{x}_{j}} \) and \( g = \mathop{\sum }\limits_{{k = 1}}^{q}{\beta }_{k}{\kappa }_{{y}_{k}} \) in \( {H}_{0} \), with \( {x}_{j},{y}_{k} \in X \) and \( {\alpha }_{j},{\beta }_{k} \in \mathbb{C} \), define \( \langle ... | Yes |
Let \( \left( {D,\mathcal{A}}\right) \) be a measurable space, where \( D \) is a nonempty set and \( \mathcal{A} \) is a \( \sigma \) -algebra on \( D \) (the measurable sets). Let \( X \) be a subset of \( \mathcal{A} \) . If \( \mu \) is a positive measure on \( \left( {D,\mathcal{A}}\right) \) and if \( \mu \) is f... | Let \( H = {\mathrm{L}}_{\mu }^{2}\left( {D,\mathcal{A},\mathbb{R}}\right) \) be the Hilbert space of \( \mu \) -square-integrable functions, with the inner product\n\n\[ \n\langle f, g\rangle = {\int }_{D}f\left( s\right) g\left( s\right) {d\mu }\left( s\right)\n\]\n\nand let \( \varphi : X \rightarrow H \) be the fea... | Yes |
Proposition 55.1. If Problem \( \left( {\mathrm{{SVM}}}_{s{2}^{\prime }}\right) \) has an optimal solution with \( w \neq 0 \) and \( \eta > 0 \) , then the following facts hold:\n\n(1) At most \( \nu \left( {p + q}\right) /2 \) points \( {u}_{i} \) fail to achieve the margin \( \eta \), and at most \( \nu \left( {p + ... | Proof. (1) Recall that for an optimal solution with \( w \neq 0 \) and \( \eta > 0 \), we have \( \gamma = 0 \), so by \( \left( { * }_{\gamma }\right) \) we have the equations\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{p}{\lambda }_{i} = \frac{{K}_{m}}{2}\;\text{ and }\;\mathop{\sum }\limits_{{j = 1}}^{q}{\mu }_{j} = \fra... | Yes |
Proposition 55.4. If Problem \( \left( {\mathrm{{SVM}}}_{s4}\right) \) has an optimal solution with \( w \neq 0 \) and \( \eta > 0 \) then the following facts hold:\n\n(1) At most \( \nu \left( {p + q}\right) \) points \( {u}_{i} \) and \( {v}_{j} \) fail to achieve the margin \( \eta \) .\n\n(2) At least \( \nu \left(... | Proof. (1) Recall that for an optimal solution with \( w \neq 0 \) and \( \eta > 0 \) we have the equation\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{p}{\lambda }_{i} + \mathop{\sum }\limits_{{j = 1}}^{q}{\mu }_{j} = \nu \]\n\nIf \( {u}_{i} \) fails to achieve the margin \( \eta \), then \( {\epsilon }_{i} > 0 \), and by c... | Yes |
1.11 Theorem Suppose \( S \) is an ordered set with the least-upper-bound property, \( B \subset S, B \) is not empty, and \( B \) is bounded below. Let \( L \) be the set of all lower bounds of \( B \) . Then\n\n\[ \alpha = \sup L \]\n\n exists in \( S \), and \( \alpha = \inf B \).\n\nIn particular, \( \inf B \) exis... | Proof Since \( B \) is bounded below, \( L \) is not empty. Since \( L \) consists of exactly those \( y \in S \) which satisfy the inequality \( y \leq x \) for every \( x \in B \), we see that every \( x \in B \) is an upper bound of \( L \) . Thus \( L \) is bounded above. Our hypothesis about \( S \) implies theref... | Yes |
3.14 Theorem Suppose \( \left\{ {s}_{n}\right\} \) is monotonic. Then \( \left\{ {s}_{n}\right\} \) converges if and only if it is bounded. | Proof Suppose \( {s}_{n} \leq {s}_{n + 1} \) (the proof is analogous in the other case). Let \( E \) be the range of \( \left\{ {s}_{n}\right\} \) . If \( \left\{ {s}_{n}\right\} \) is bounded, let \( s \) be the least upper bound of \( E \) . Then\n\n\[ \n{s}_{n} \leq s\;\left( {n = 1,2,3,\ldots }\right) .\n\]\n\nFor ... | Yes |
Consider two partitions of \( \Omega \) of the form\n\n\[ \n{\mathfrak{c}}_{1} = \left\{ {{\omega }_{1},{\omega }_{2}}\right\} ,\;{\mathfrak{c}}_{2} = \left\{ {{\omega }_{3},{\omega }_{4}}\right\} ,\;{\mathfrak{c}}_{3} = \left\{ {{\omega }_{5},{\omega }_{6}}\right\} \n\]\n\n\[ \n{\mathfrak{d}}_{1} = \left\{ {{\omega }_... | Now we can define a finite probability space \( \{ \Omega ,\mathfrak{U}, P\} \) which is a triple consisting of the finite space of elementary outcomes \( \Omega \), the algebra of all possible outcomes \( \mathfrak{U} \) (which is the set of all subsets of \( \Omega \) ), and the probability measure \( P \) on \( \mat... | No |
Consider the random variable \( \xi \) on the space \( \Omega \) of the form\n\n\[ \xi \left( {\omega }_{k}\right) = \left| {k - 3}\right| \]\n\nThis variable assumes four distinct values \( 0,1,2,3 \) with probabilities \( 1/6,1/3 \), \( 1/3,1/6 \), respectively. Its cumulative probability distribution is\n\n\[ {F}_{\... | Accordingly,\n\n\[ \mathbb{E}\{ \xi \} = \frac{3}{2},\;\sigma \{ \xi \} = \sqrt{\frac{11}{12}},\;{\gamma }_{3}\{ \xi \} = 0,\;{\gamma }_{4}\{ \xi \} = - \frac{114}{121}. \]\n\nThe variable \( \xi \) is bimodal. Its modes are \( 1,2 \) . | Yes |
The standard normal (or Gaussian) variable \( \xi \) takes values on the entire real line. It has the c.d.f. \( {F}_{\xi }\left( x\right) \) and p.d.f. \( {f}_{\xi }\left( x\right) \) of the form\n\n\[ \n{F}_{\xi }\left( x\right) = {\int }_{-\infty }^{x}\frac{{e}^{-{\left( {x}^{\prime }\right) }^{2}/2}}{\sqrt{2\pi }}d{... | Straightforward calculation yields\n\n\[ \n{\mathbb{M}}_{k}\{ \xi \} = {\int }_{-\infty }^{\infty }{x}^{k}\frac{{e}^{-{x}^{2}/2}}{\sqrt{2\pi }}{dx} = \left\{ \begin{array}{ll} 0, & k\text{ is odd } \\ 1 \cdot 3\ldots \cdot \left( {k - 1}\right) , & k\text{ is even } \end{array}\right.\n\]\n\nAccordingly,\n\n\[ \n\mathb... | Yes |
The Poisson distribution is a discrete distribution of the form\n\n\[ P\left( {\xi = k}\right) = \left\{ \begin{matrix} 0,\;k < 0 \\ {e}^{-\theta }{\theta }^{k}/k!,\;k \geq 0 \end{matrix}\right. \]\n\nwhere \( \theta > 0 \) is a free parameter and \( k \) is an integer. | A simple calculation shows that\n\n\[ \mathbb{E}\{ \xi \} = \theta ,\;\sigma \{ \xi \} = \sqrt{\theta }, \]\n\n\[ \begin{matrix} {\gamma }_{3}\left\{ \xi \right\} & = & 1/\sqrt{\theta },\;{\gamma }_{4}\left\{ \xi \right\} = 1/\theta . \end{matrix} \] | Yes |
The most important example of the vector random variable is the standard \( N \) -component normal variable \( \xi = \left( {{\xi }^{1},\ldots ,{\xi }^{N}}\right) \), where \( {\xi }^{n} \) are independent normal variables. We denote this variable by \( \xi \sim \mathfrak{N}\left( {\mathbf{0},\mathcal{I}}\right) \), wh... | \[ {f}_{\mathbf{\xi }}\left( \mathbf{x}\right) = \frac{{e}^{-{\left| \mathbf{x}\right| }^{2}/2}}{\sqrt{{\left( 2\pi \right) }^{N}}} \] | Yes |
For the random variable \( \xi \) and partition \( {\mathfrak{c}}_{1},{\mathfrak{c}}_{2},{\mathfrak{c}}_{3} \), we have\n\n\[ \mathbb{E}\left\{ {\xi \mid {\mathfrak{U}}_{\mathfrak{c}}}\right\} \left( {\omega }_{k}\right) = \left\{ \begin{array}{ll} 3/2, & k = 1,2 \\ 1/2, & k = 3,4 \\ 5/2, & k = 5,6 \end{array}\right. \... | \[ \mathbb{E}\left\{ {\mathbb{E}\left\{ {\xi \mid {\mathfrak{U}}_{\mathrm{c}}}\right\} }\right\} = \frac{3}{2} \cdot \frac{1}{3} + \frac{1}{2} \cdot \frac{1}{3} + \frac{5}{2} \cdot \frac{1}{3} = \frac{3}{2} = \mathbb{E}\{ \xi \} . \] | Yes |
It is easy to find the characteristic function for the normal distribution by completing the squares: | \[ \Phi \left( \varsigma \right) = \int \frac{{e}^{i\varsigma x - {x}^{2}/2}}{\sqrt{2\pi }}{dx} = {e}^{-{\varsigma }^{2}/2}. \] | Yes |
The principal example of a Markov process with continuous trajectories is the ubiquitous scalar Wiener process \( {W}_{t} \) . This process is characterized by its nonrandom initial value \( {W}_{0} = 0 \) and the stationary t.p.d.f. | It is easy to verify that this t.p.d.f. satisfies the Chapman-Kolmogoroff equation. Moreover, by using Kolmogoroff's condition (4.2), one can show that trajectories of \( {W}_{t} \) are continuous. As we will see shortly, these trajectories are nowhere differentiable. For fixed \( t \) the scalar random variable \( {W}... | No |
Example 4.2. The principal example of a vector Markov process with continuous trajectories is the standard \( N \) -component Wiener processes \( {W}_{t} = \) \( \left( {{W}_{t}^{1},\ldots ,{W}_{t}^{N}}\right) \), where \( {W}_{t}^{n} \) are independent scalar Wiener processes. The corresponding t.p.d.f. is | \[ p\left( {\tau ,\mathbf{x},{\mathbf{x}}^{\prime }}\right) = \frac{{e}^{-{\left| {\mathbf{x}}^{\prime } - \mathbf{x}\right| }^{2}/{2\tau }}}{{\left( 2\pi \tau \right) }^{N/2}} = H\left( {\tau ,{\mathbf{x}}^{\prime } - \mathbf{x}}\right) . \] (4.6) For fixed \( t \) the vector random variable \( {W}_{t} \) has an \( N ... | Yes |
In this example we treat the scalar Wiener process as a diffusion process. Since this process is homogeneous, we know that \( p = p\left( {\tau, x,{x}^{\prime }}\right) \) , \( a = a\left( x\right), b = b\left( x\right) \) . The killing rate is zero, \( c\left( x\right) = 0 \) . Straightforward calculation which is lef... | \[ {p}_{\tau } - \frac{1}{2}{p}_{xx} = 0 \] (4.21) supplied with the initial condition \[ p\left( {0, x,{x}^{\prime }}\right) = \delta \left( {x - {x}^{\prime }}\right) . \] The corresponding natural scale density, speed and killing measures are constant, \( \mathrm{s}\left( x\right) = 1,\mathrm{\;m}\left( x\right) = 1... | No |
We consider the standard Wiener process on the semi-infinite interval \( \left( {0,\infty }\right) \). We already know that the Kolmogoroff and Fokker-Planck equations have the same form (4.21). Since we consider these equations on the half-axis, we have to augment them with appropriate boundary conditions. By definiti... | In the first case the process is immediately returned back to the interval \( \left( {0,\infty }\right) \) , in the second case it is terminated as soon as it reaches zero, finally, in the third case a combination of the above occurs. It is easy to find the corresponding t.p.d.f.’s in the first and second case by the m... | Yes |
Example 4.5. We consider the so-called Bessel process with\n\n\[ \na\left( x\right) = 1,\;b\left( x\right) = \frac{\left( 1 - 2\nu \right) }{2x},\;c\left( x\right) = 0.\n\]\n\nIt is natural to consider such a process on the semi-infinite interval \( \left( {0,\infty }\right) \) . The corresponding Kolmogoroff equation ... | The expression in parenthesis is the radial part of the Laplace operator in \( 2\left( {1 - \nu }\right) \) dimensional space. The natural scale density and the speed measure are\n\n\[ \n\mathrm{s}\left( x\right) = {x}^{-\left( {1 - {2\nu }}\right) },\;\mathrm{m}\left( x\right) = {x}^{\left( 1 - 2\nu \right) }.\n\]\n\n... | Yes |
If we choose \( {x}_{t} = {W}_{t} \) and \( f\left( {t, x}\right) = {x}^{2}/2 \), we get\n\n\[ d\left\lbrack {\frac{1}{2}{W}_{t}^{2}}\right\rbrack = \frac{1}{2}{dt} + {W}_{t}d{W}_{t} \] | so that\n\n\[ {I}_{0, t} = {\int }_{0}^{t}{W}_{s}d{W}_{s} = \frac{1}{2}{W}_{t}^{2} - \frac{1}{2}t \]\n\nin agreement with formula (4.42). | Yes |
Example 5.1. We consider a two-country market with nonrisky investments over a period of one year. The states of this market are described by the following table\n\n<table><thead><tr><th></th><th>\\( p \\)</th><th>\\( {r}^{0} \\)</th><th>\\( {r}^{1} \\)</th><th>\\( {S}_{0}^{01} \\)</th><th>\\( {S}_{1}^{01} \\)</th></tr... | <table><thead><tr><th></th><th>\\( {\\widehat{\\varrho }}^{01} \\)</th><th>\\( {q}^{0} \\)</th><th>\\( {q}^{1} \\)</th><th>\\( {N}^{10} \\)</th></tr></thead><tr><td>\\( {\\omega }_{0} \\)</td><td>\\( - {0.128} \\)</td><td>0.413</td><td>0.361</td><td>0.872</td></tr><tr><td>\\( {\\omega }_{1} \\)</td><td>0.090</td><td>0.... | Yes |
We now consider the market introduced in Example 5.1 and evaluate all the derivative instruments discussed in this section and find the corresponding hedges. Specifically we consider the domestic bond, forward contract, and ITMF call and OTMF put with strike \( K = {0.85} \). In addition we check put-call parity. | <table><thead><tr><th></th><th>\( {B}^{0} \)</th><th>\( F \)</th><th>\( C \)</th><th>\( P \)</th><th>\( C - P \)</th><th>\( {B}^{0}\left( {F - K}\right) \)</th></tr></thead><tr><td>\( {V}_{0} \)</td><td>0.943</td><td>0.917</td><td>\( {0.083} \)</td><td>\( {0.020} \)</td><td>0.063</td><td>0.063</td></tr><tr><td>\( {\Del... | Yes |
Now we show how to price and hedge all the relevant contingent claims in the market introduced above. We consider the domestic bond, forward contract, and call and put with strike \( K = {0.85} \) and obtain | <table><thead><tr><th></th><th>\( {B}^{0} \)</th><th>\( F \)</th><th>\( {C}^{0} \)</th><th>\( {P}^{0} \)</th></tr></thead><tr><td>\( {V}_{0} \)</td><td>0.941</td><td>0.915</td><td>0.081</td><td>0.020</td></tr><tr><td>\( {\Delta }^{0} \)</td><td>1.103</td><td>\( - {0.861} \)</td><td>-0.554</td><td>0.240</td></tr><tr><td... | Yes |
In order to illustrate the ideas developed above we revisit the binomial two-country market studied in Section 5.3. The matrix of relative FXRs \( {Z}_{1} \), and its transpose \( {Z}_{1}^{T} \) have the form\n\n\[ \n{\mathcal{Z}}_{1} = \left( \begin{matrix} 1 & 1 \\ {\widehat{A}}_{1,0}^{01} & {\widehat{A}}_{1,1}^{01} ... | Thus, any nondegenerate market obeys the LOP. The NSDSC is satisfied provided that\n\n\[ \n{\widehat{A}}_{1,0}^{01} \leq {S}_{0}^{01} \leq {\widehat{A}}_{1,1}^{01} \n\]\n\nor, equivalently, if\n\n\[ \n{S}_{1,0}^{01} \leq {F}^{01} \leq {S}_{1,1}^{01} \n\]\n\nThe NAC is true provided that these inequalities are strict. F... | Yes |
Example 5.6. (The trinomial two-country market.) In this example we consider the trinomial two-country market which can end up in three different states. We can represent the corresponding matrices \( {\mathcal{Z}}_{1},{\mathcal{Z}}_{1}^{T} \) as\n\n\[ \n{\mathcal{Z}}_{1} = \left( \begin{matrix} 1 & 1 & 1 \\ {\widehat{... | It is easy to check that the solution of equations (5.19) can be written in the vector form as follows\n\n\[ \nq = \frac{\widehat{\varrho } \cdot \widehat{\varrho }E - \widehat{\varrho } \cdot E\widehat{\varrho }}{\widehat{\varrho } \cdot \widehat{\varrho }E \cdot E - {\left( \widehat{\varrho } \cdot E\right) }^{2}} + ... | Yes |
We can use the above formulas in order to price and hedge a European call with strike \( K = {0.85} \) in our three-period model. | <table><thead><tr><th></th><th>\( {C}_{0} \)</th><th>\( {\Delta }_{0}^{1} \)</th><th>\( {C}_{1} \)</th><th>\( {\Delta }_{1}^{1} \)</th><th>\( {C}_{2} \)</th><th>\( {\Delta }_{2}^{1} \)</th><th>\( {C}_{3} \)</th></tr></thead><tr><td>\( {\omega }_{0} \)</td><td>0.162</td><td>0.609</td><td>\( {0.055} \)</td><td>0.376</td>... | Yes |
To illustrate the above ideas, we price and hedge a floating strike lookback call in our three-period model. | <table><thead><tr><th></th><th>\\( {V}_{0}^{\\left( LBC\\right) } \\)</th><th>\\( {\\Delta }_{0}^{1} \\)</th><th>\\( {V}_{1}^{\\left( LBC\\right) } \\)</th><th>\\( {\\Delta }_{1}^{1} \\)</th><th>\\( {V}_{2}^{\\left( LBC\\right) } \\)</th><th>\\( {\\Delta }_{2}^{1} \\)</th><th>\\( {V}_{3}^{\\left( LBC\\right) } \\)</th>... | Yes |
Example 6.5. At this stage it is useful to consider a representative three-period market. Let us assume that this market is constructed by combining three different single-period markets with deterministic interest rates and volatilities given by the table\n\n<table><thead><tr><th></th><th>\\( {r}^{0} \\)</th><th>\\( {... | The tree describing possible states of the market is shown in Figure 6.4. This tree, which is obviously nonrecombining, is more complicated than the recombining tree shown in Figure 6.1. In particular, the terminal FXR can have \\( 8 = {2}^{3} \\) distinct values \\( {S}_{3}\\left( {\\omega }_{l}\\right), l = 0,...,7 \... | Yes |
Example 6.6. We calibrate a three-period recombining market with the spot FXR \( {S}_{0} = {0.9} \) and constant interest rates \( {r}^{0} = {0.06},{r}^{1} = {0.04} \), to market prices of calls given by the following table | Here \( \varkappa \) is the moneyness of the corresponding call which is defined as follows: \( \varkappa = {F}_{0, t}/K \) . Thus, calls with \( \varkappa > 1,\varkappa = 1 \), and \( \varkappa < 1 \) are ITMF, ATMF, and OTMF, respectively. We have \( {F}_{0,1,0} = {0.917} \) . In order to find \( {S}_{1, \pm 1} \) we... | No |
Consider a flat boundary one-touch option. It is clear that for such an option\n\n\[ a\left( \chi \right) = 0,\;\eta \left( \chi \right) = {e}^{\lambda \chi } \]\n\nwhere\n\n\[ \lambda = \left\{ \begin{array}{ll} {\widehat{r}}^{0} + {\widehat{\gamma }}_{ - }^{2}/2, & \text{ (immediate rebate) } \\ {\widehat{\gamma }}_{... | In the case in question the integral equations becomes trivial and yields\n\n\[ \nu \left( \chi \right) = \eta \left( \chi \right) = {e}^{\lambda \chi } \]\n\nThe integral representation for the solution assumes the form\n\n\[ U\left( {\chi, X}\right) = {\int }_{0}^{\chi }\frac{X{e}^{-{X}^{2}/2\left( {\chi - {\chi }^{\... | Yes |
Consider now the one-touch option with an exponential boundary. The nondimensional boundary is straight,\n\n\\[ \n a\\left( \\chi \\right) = {\\alpha \\chi } \n\\]\n\nThe pricing problem has the form (12.10) - (12.12). The integral equation for \\( \\nu \\left( \\chi \\right) \\) assumes the form\n\n\\[ \n \\nu \\left(... | It is an equation in convolutions which can be solved via the Laplace transforms. The usual sequence of operations which are left to the reader as an exercise yields the following expression for the Laplace transformed density,\n\n\\[ \n \\nu \\left( s\\right) = \\frac{\\sqrt{s + {\\alpha }^{2}/2}}{\\left( {s - \\mu }\... | No |
Corollary 4.14. For continued fractions \( \mathrm{K}\left( {1/{b}_{n}}\right) \) let\n\n\[ \n{V}_{n} = \left\lbrack {w : \left| {w - c}\right| \leq \sqrt{1 + {c}^{2}}}\right\rbrack ,\;\text{ creal,}\;n = 0,1,2,\ldots ,\n\]\n\n(4.2.37)\n\n\[ \n{G}_{n} = \left\lbrack {w : \left| {w + {2c}}\right| \geq 2\sqrt{1 + {c}^{2}... | Although the preceding is indeed a corollary of Theorem 4.3, the computation is extremely tedious. The result was originally obtained in a different manner (see [Thron, 1944a]) and would have been difficult to deduce from Theorem 4.3. The following, however, can be obtained without difficulty. | No |
Corollary 4.20. The continued fraction \( \mathrm{K}\left( {{a}_{n}/{b}_{n}}\right) \) diverges if \( \sum \left| {b}_{n}^{ * }\right| < \infty \) , where\n\n\[ \n{b}_{{2n} - 1}^{ * } = {b}_{{2n} - 1}\frac{{a}_{2}{a}_{4}\cdots {a}_{{2n} - 2}}{{a}_{1}{a}_{3}\cdots {a}_{{2n} - 1}},\;n = 1,2,3,\ldots ,\n\]\n\n(4.3.4a)\n\n... | Proof. \( \mathrm{K}\left( {{a}_{n}/{b}_{n}}\right) \) is equivalent to \( \mathrm{K}\left( {1/{b}_{n}^{ * }}\right) \) [see (2.3.23)]. | No |
Lemma 4.23. If\n\n\\[ \n{s}_{n}\left( w\right) = \frac{{a}_{n}}{1 + w},\;{a}_{n} \neq 0,\;n = 1,2,\ldots, m, \n\\]\n\nand\n\n\\[ \n- 1 = {s}_{1} \circ {s}_{2} \circ \cdots \circ {s}_{m}\left( 0\right) , \n\\]\n\n(4.3.6)\n\nthen\n\n\\[ \n- 1 = {s}_{m} \circ {s}_{m - 1} \circ \cdots \circ {s}_{1}\left( 0\right) . \n\\]\n... | Proof. We recall from (4.2.16) and (4.2.18) that if \( v\left( w\right) = - 1 - w \), then\n\n\\[ \n{s}_{n}\left( w\right) = v \circ {s}_{n}^{-1} \circ v\left( w\right) \n\\]\n\n(4.3.8)\n\nand \( v \) is an idempotent l.f.t. The hypothesis (4.3.6) can be written as \( v\left( 0\right) = {s}_{1} \circ {s}_{2} \circ \cdo... | Yes |
Corollary 4.36. (A) \( \mathrm{K}\left( {{a}_{n}/{b}_{n}}\right) \) converges to a finite value if\n\n\[ \left| \frac{{a}_{1}}{{b}_{1}}\right| \leq \frac{{p}_{1} - 1}{{p}_{1}} \]\n\n(4.4.19a)\n\n\[ \frac{\left| {a}_{n}\right| }{\left| {b}_{n - 1}{b}_{n}\right| } \leq \frac{{p}_{n} - 1}{{p}_{n - 1}{p}_{n}},\;n = 2,3,4,\... | Proof. (A) is obtained by an application of Theorem 4.35 to the equivalent continued fraction\n\n\[ \frac{{\rho }_{1}{a}_{1}}{{p}_{1}{b}_{1}} + \frac{{\rho }_{1}{\rho }_{2}{a}_{2}}{{p}_{2}{b}_{2}} + \frac{{\rho }_{2}{\rho }_{3}{a}_{3}}{{p}_{3}{b}_{3}} + \cdots \]\n\nand then setting \( {\rho }_{n} = {p}_{n}/{b}_{n} \) ... | Yes |
Property 2. If \( D\left( {a, f}\right) > k/2 \), then the minimum of \( \left\lbrack {D\left( {a, q}\right) : q \in P}\right\rbrack \) is attained at exactly two points \( {q}_{1} \) and \( {q}_{2} \) on \( P \), which are symmetric with respect to the axis of \( P \) . Moreover, | \[ D\left( {f,{q}_{0}}\right) = D\left( {f, a}\right) ,\;{q}_{0} \in P, \] iff \[ D\left( {f,{q}_{0}}\right) = \mathop{\min }\limits_{{q \in P}}D\left( {a, q}\right) . \] To verify these properties it will suffice to consider a parabola \( P \) with focus \( f \) at the origin, axis along the real axis and vertex \( v ... | Yes |
Lemma 4.47. If \( a > 0 \) and \( 0 \leq k < 1 \), then\n\n\[ H = \left| {1 + \frac{a{e}^{i\alpha }}{1 + k{e}^{i\varphi }}}\right| \geq \frac{\left| {a}^{2} + 2a\cos \alpha + 1 - {k}^{2}\right| }{\left| {1 - {k}^{2} + a{e}^{i\alpha }}\right| + {ka}}, \]\n\nwhere \( \alpha \) and \( \varphi \) are arbitrary real numbers... | Proof. The set \( \left\lbrack {w : w = 1/\left( {1 + k{e}^{i\varphi }}\right) ,0 \leq \varphi \leq {2\pi }}\right\rbrack \) is the image of the circle \( \left\lbrack {z : \left| {z - 1}\right| = k}\right\rbrack \) under the mapping \( w = 1/z \) . Hence the image set is a circle and, by an argument analogous to the o... | Yes |
Given a sequence \( \left\{ {{R}_{n}\left( z\right) }\right\} \) of functions meromorphic at the origin, there exists an fLs \( L \) such that \( \left\{ {{R}_{n}\left( z\right) }\right\} \) corresponds to \( L \) iff\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\lambda \left( {L\left( {R}_{n + 1}\right) - L\le... | In view of the preceding discussion, to prove (A) it suffices to show that \( \left\{ {L\left( {R}_{n}\right) }\right\} \) is a Cauchy sequence with respect to the metric \( \rho \) [see (5.1.15)] iff (5.1.18) holds. We note that condition (5.1.17) is equivalent to\n\n\[ \lambda \left( {L\left( {R}_{n + k}\right) - L\l... | Yes |
Every C-fraction\n\n\[ 1 + \frac{{a}_{1}{z}^{{\alpha }_{1}}}{1} + \frac{{a}_{2}{z}^{{\alpha }_{2}}}{1} + \frac{{a}_{3}{z}^{{\alpha }_{3}}}{1} + \cdots ,\;{a}_{n} \neq 0, \]\n\nwhere the \( {\alpha }_{n} \) are positive integers, corresponds to a uniquely determined formal power series (fps) of the form\n\n\[ {L}_{0} = ... | Proof. (A): If \( {A}_{n}\left( z\right) \) and \( {B}_{n}\left( z\right) \) denote the \( n \) th numerator and denominator, respectively, of the \( C \) -fraction (5.1.42), then from the difference equations (2.1.6) one sees that \( {A}_{n}\left( z\right) \) and \( {B}_{n}\left( z\right) \) are polynomials in \( z \)... | Yes |
\[ \arctan z = {zF}\left( {\frac{1}{2},1;\frac{3}{2}; - {z}^{2}}\right) \] | \[ = \frac{z}{1} + \frac{{1}^{2}{z}^{2}}{3} + \frac{{2}^{2}{z}^{2}}{5} + \frac{{3}^{2}{z}^{2}}{7} + \frac{{4}^{2}{z}^{2}}{9} + \cdots ,\] (6.1.14) as given by (2.1.16). The continued fraction converges and represents a single-valued branch of the analytic function arctan \( z \) in the cut \( z \) -plane with cuts alon... | Yes |
\[ {\left( 1 + z\right) }^{\alpha } = F\left( {-\alpha ,1;1; - z}\right) \] | \[ = \frac{1}{1} + \frac{\left( {-\alpha }\right) z}{1} + \frac{1\left( {1 + \alpha }\right) z}{2} + \frac{1\left( {1 - \alpha }\right) z}{3} + \frac{2\left( {2 + \alpha }\right) z}{4} + \frac{2\left( {2 - \alpha }\right) z}{5} + \cdots ,\] | No |
\[ \log \left( {1 + z}\right) = {zF}\left( {1,1;2; - z}\right) \] | \[ = \frac{z}{1} + \frac{{1}^{2}z}{2} + \frac{{1}^{2}z}{3} + \frac{{2}^{2}z}{4} + \frac{{2}^{2}z}{5} + \frac{{3}^{2}z}{7} + \frac{{3}^{2}z}{9} + \cdots \text{.} \] | No |
\[ {\int }_{0}^{z}\frac{dt}{1 + {t}^{n}}{dt} = {zF}\left( {\frac{1}{n},1;1 + \frac{1}{n}; - {z}^{n}}\right) \] | \[ = \frac{z}{1} + \frac{{1}^{2} \cdot {z}^{n}}{n + 1} + \frac{{\left( 1 \cdot n\right) }^{2}{z}^{n}}{{2n} + 1} + \frac{{\left( n + 1\right) }^{2}{z}^{n}}{{3n} + 1} \] \[ + \frac{{\left( 2n\right) }^{2}{z}^{n}}{{4n} + 1} + \frac{{\left( 2n + 1\right) }^{2}{z}^{n}}{{5n} + 1} + \cdots , \] \[ n \in \left\lbrack {1,2,3,\l... | Yes |
By Theorem 6.1 we obtain\n\n\[ \frac{\arcsin z}{\sqrt{1 - {z}^{2}}} = \frac{{zF}\left( {\frac{1}{2},\frac{1}{2};\frac{3}{2};{z}^{2}}\right) }{F\left( {\frac{1}{2}, - \frac{1}{2};\frac{1}{2};{z}^{2}}\right) } \] | \[ = \frac{z}{1} - \frac{1 \times 2{z}^{2}}{3} - \frac{1 \times 2{z}^{2}}{5} - \frac{3 \times 4{z}^{2}}{7} - \frac{3 \times 4{z}^{2}}{9} - \frac{5 \times 6{z}^{2}}{11} - \frac{5 \times 6{z}^{2}}{13} - \cdots ,\] \n\n(6.1.20)\n\nwhere the continued fraction converges and represents a single-valued branch of the analytic... | Yes |
\[ {e}^{z} = \Phi \left( {1;1;z}\right) \] | \[ = \frac{1}{1} - \frac{z}{1} + \frac{1 \cdot z}{2} - \frac{1 \cdot z}{3} + \frac{2 \cdot z}{4} - \frac{2 \cdot z}{5} + \frac{3 \cdot z}{6} - \frac{3 \cdot z}{7} + \cdots ,\] \n\n(6.1.35) \n\nis valid for all \( z \in \mathbb{C} \) . It is easily seen that (6.1.35) is equivalent to \n\n\[ {e}^{z} = \frac{1}{1} - \frac... | Yes |
Example 8. Error function.\n\n\\[ \n\\operatorname{erf}z = \\frac{2}{\\sqrt{\\pi }}{\\int }_{0}^{z}{e}^{-{t}^{2}}{dt} = \\frac{2z}{\\sqrt{\\pi }}{e}^{-{z}^{2}}\\Phi \\left( {1;\\frac{3}{2};{z}^{2}}\\right) \n\\] | \\[ \n= \\frac{2}{\\sqrt{\\pi }}{e}^{-{z}^{2}}\\left( {\\frac{z}{1} - \\frac{{z}^{2}}{\\frac{3}{2}} + \\frac{1 \\cdot {z}^{2}}{\\frac{5}{2}} - \\frac{\\frac{3}{2} \\cdot {z}^{2}}{\\frac{7}{2}} + \\frac{2 \\cdot {z}^{2}}{\\frac{9}{2}} - \\frac{\\frac{5}{2} \\cdot {z}^{2}}{\\frac{11}{2}} + \\frac{3 \\cdot {z}^{2}}{\\frac... | Yes |
Fresnel integrals are defined by\n\n\[ C\left( z\right) = {\int }_{0}^{z}\cos \left( {\frac{\pi }{2}{t}^{2}}\right) {dt},\;S\left( z\right) = {\int }_{0}^{z}\sin \left( {\frac{\pi }{2}{t}^{2}}\right) {dt}. \] | These functions are related to the error function and confluent hypergeometric functions by\n\n\[ C\left( z\right) + {iS}\left( z\right) = \frac{1 + i}{2}\operatorname{erf}\left( {\frac{\sqrt{\pi }}{2}\left( {1 - i}\right) z}\right) = z{e}^{\left( {{i\pi }/2}\right) {z}^{2}}\Phi \left( {1;\frac{3}{2}; - \frac{i\pi }{2}... | Yes |
Incomplete gamma functions are defined by\n\n\[ \gamma \left( {a, z}\right) = {\int }_{0}^{z}{e}^{-t}{t}^{a - 1}{dt},\;\operatorname{Re}a > 0. \] | These functions are related to confluent hypergeometric functions by\n\n\[ \gamma \left( {a, z}\right) = {a}^{-1}{z}^{a}{e}^{-z}\Phi \left( {1;a + 1;z}\right) . \]\n\nThus by (6.1.33)\n\n\[ \gamma \left( {a, z}\right) = {a}^{-1}{z}^{a}{e}^{-z}\left( {\frac{1}{1} - \frac{z}{a + 1} + \frac{1 \cdot z}{a + 2} - \frac{\left... | Yes |
The Bessel function \( {J}_{\nu }\left( z\right) \) of the first kind of order \( \nu \) can be expressed by \[ {J}_{\nu }\left( z\right) = \frac{{\left( z/2\right) }^{\nu }}{\Gamma \left( {\nu + 1}\right) }\Psi \left( {\nu + 1; - \frac{{z}^{2}}{4}}\right) ,\;z \in \mathbb{C}, \] provided \( \nu \notin \left\lbrack {-1... | Thus \[ \frac{{J}_{\nu }\left( z\right) }{{J}_{\nu + 1}\left( z\right) } = \frac{2\left( {\nu + 1}\right) }{z}\frac{\Psi \left( {\nu + 1; - {z}^{2}/4}\right) }{\Psi \left( {\nu + 2; - {z}^{2}/4}\right) }, \] and hence, by Theorem 6.4 and an equivalence transformation, we obtain \[ \frac{{J}_{\nu + 1}\left( z\right) }{{... | Yes |
Example 12. Tangent function. Since\n\n\\[ \n\\sin z = {z\\Psi }\\left( {\\frac{3}{2}; - \\frac{{z}^{2}}{4}}\\right) \\text{ and }\\cos z = \\Psi \\left( {\\frac{1}{2}; - \\frac{{z}^{2}}{4}}\\right) ,\n\\]\n\nwe obtain from Theorem 6.4 and an equivalence transformation\n\n\\[ \n\\tan z = z\\frac{\\Psi \\left( {\\frac{3... | \\[ \n= \\frac{z}{1} - \\frac{{z}^{2}}{3} - \\frac{{z}^{2}}{5} - \\frac{{z}^{2}}{7} - \\frac{{z}^{2}}{9} - \\cdots \\text{. }\n\\]\n\n(6.1.55)\n\nThe continued fraction in (6.1.55) converges and represents \\( \\tan z \\) for all \\( z \\in \\mathbb{C} \\) . Replacing \\( z \\) by \\( {iz} \\) in (6.1.55) gives\n\n\\[ ... | Yes |
Legendre functions of the first kind of degree \( \alpha \) and order \( m \) are defined by\n\n\[ \n{P}_{\alpha }^{m}\left( z\right) = \frac{1}{\Gamma \left( {1 - m}\right) }{\left( \frac{z + 1}{z - 1}\right) }^{m/2}F\left( {-\alpha ,\alpha + 1;1 - m;\frac{1}{2} - \frac{1}{2}z}\right) ,\;\left| {1 - z}\right| < 2 \n\]... | Let \( z \) and \( \alpha \) be fixed, \( \alpha \) not an integer, and define \( \left\{ {y}_{m}\right\} \) by\n\n\[ \n{y}_{m} = {P}_{\alpha }^{m}\left( z\right) ,\;m = 0,1,2,\ldots \n\] \n\nGautschi [1967, Section 6] has shown that, for \( \operatorname{Re}\left( z\right) > 0,\left\{ {y}_{m}\right\} \) is a minimal s... | Yes |
Coulomb wave functions play an important role in the study of nuclear interactions. The regular Coulomb wave function \( {F}_{L}\left( {\eta ,\rho }\right) \) is defined\n\n\[ \n{F}_{L}\left( {\eta ,\rho }\right) = {\rho }^{L + 1}{e}^{-{i\rho }}{C}_{L}\left( \eta \right) \Phi \left( {L + 1 - {i\eta};{2L} + 2;{2i\rho }}... | Gautschi [1967, Section 7] has shown that for fixed \( \eta \) and \( \rho ,\left\{ {{F}_{L}\left( {\eta ,\rho }\right) }\right\} \) is a minimal solution of the system of three-term recurrence relations\n\n\[ \nL{\left\lbrack {\left( L + 1\right) }^{2} + {\eta }^{2}\right\rbrack }^{1/2}{y}_{L + 1} = \left( {{2L} + 1}\... | Yes |
Incomplete beta functions are defined by\n\n\\[ \n{B}_{x}\left( {p, q}\right) = {\int }_{0}^{x}{t}^{p - 1}{\left( 1 - t\right) }^{q - 1}{dt},\;p > 0,\;q > 0,\;0 \leq x \leq 1. \n\\] | It can be shown that\n\n\\[ \n{B}_{x}\left( {p, q}\right) = {p}^{-1}{x}^{p}F\left( {p,1 - q;p + 1;x}\right) , \n\\]\n\nwhere \\( F \\) denotes the hypergeometric function (6.1.1),(see [Erdelyi et al., 1953, Vol. 1, p. 87]). | Yes |
Incomplete gamma function. We consider here the incomplete gamma function\n\n\[ P\\left( {a, x}\\right) = \\frac{1}{\\Gamma \\left( a\\right) }{\\int }_{0}^{x}{e}^{-t}{t}^{a - 1}{dt} \]\n\n(6.2.15a)\n\nwith the restrictions\n\n\[ a > 0,\;x > 0. \]\n\n(6.2.15b)\n\nWe note that \( \\Gamma \\left( a\\right) P\\left( {a, x... | Gautschi [1967, Section 8] has shown that \( \\left\{ {h}_{n}\\right\} \) is a minimal solution of the system of three-term recurrence formulas\n\n\[ {y}_{n + 1} = \\left( {1 + \\frac{x}{a + n}}\\right) {y}_{n} - \\frac{x}{a + n}{y}_{n - 1},\\;n = 1,2,3,\\ldots \]\n\n(6.2.17)\n\nHence by Theorem 5.7 (Pincherle), for ea... | Yes |
Repeated integrals of the error function. The complementary error function \( \operatorname{erfc}z \) is defined by\n\n\[ \operatorname{erfc}z = \frac{2}{\sqrt{\pi }}{\int }_{z}^{\infty }{e}^{-{t}^{2}}{dt} = 1 - \operatorname{erf}z \]\n\n(6.2.19)\n\n[see (6.1.38)]. Here the path of integration is subject to the restric... | Further let \( \left\{ {h}_{n}\right\} \) and \( \left\{ {g}_{n}\right\} \) be defined by\n\n\[ {h}_{n} = {e}^{{z}^{2}}{I}^{n}\operatorname{erfc}z,\;n = - 1,0,1,2,\ldots , \]\n\n(6.2.21a)\n\n\[ {g}_{n} = {\left( -1\right) }^{n}{e}^{{z}^{2}}{I}^{n}\operatorname{erfc}\left( {-z}\right) ,\;n = - 1,0,1,2,\ldots . \]\n\n(6.... | Yes |
Every regular C-fraction (7.1.1) corresponds to a uniquely determined fps\n\n\[ L = 1 + {c}_{1}z + {c}_{2}{z}^{2} + {c}_{3}{z}^{3} + \cdots . \]\n\n(7.1.2)\n\nThe order of correspondence of the \( n \) th approximant \( {f}_{n}\left( z\right) \) is \( {\nu }_{n} = n + 1 \), so that the Taylor expansion of \( {f}_{n}\le... | Part (A) is an immediate consequence of Corollary 5.3 (A). To prove (B) we let \( {A}_{n}\left( z\right) \) and \( {B}_{n}\left( z\right) \) denote the \( n \) th numerator and denominator, respectively, of (7.1.1), and let \( {A}_{n}^{ \dagger }\left( z\right) \) and \( {B}_{n}^{ \dagger }\left( z\right) \) denote the... | Yes |
Arctangent. The fps\n\n\[ L = 1 + z - \frac{{z}^{2}}{3} + \frac{{z}^{3}}{5} - \frac{{z}^{4}}{7} + \frac{{z}^{5}}{9} - \cdots \]\n\n(7.1.45)\n\nconverges to the function\n\n\[ f\left( z\right) = 1 + {z}^{1/2}\arctan {z}^{1/2} \]\n\n(7.1.46)\n\nat least for \( \left| z\right| \leq 1, z \neq - 1 \) . | The initial part of the qd table for \( L \) is given in Table 7.1.2. From this (and Theorem 7.6) we can see that the regular C-fraction\n\n\[ 1 + \frac{z}{1} + \frac{\frac{1}{3}z}{1} + \frac{\frac{{2}^{2}}{3 \times 5}z}{1} + \frac{\frac{{3}^{2}}{5 \times 7}z}{1} + \frac{\frac{{4}^{2}}{7 \times 9}z}{1} + \cdots \]\n\n(... | Yes |
For the series\n\n\[ F\left( {a,1;c;z}\right) = 1 + \frac{a}{c}z + \frac{a\left( {a + 1}\right) }{c\left( {c + 1}\right) }{z}^{2} + \frac{a\left( {a + 1}\right) \left( {a + 2}\right) }{c\left( {c + 1}\right) \left( {c + 2}\right) }{z}^{3} + \cdots ,\ ]\n\n(7.1.49)\n\nwith \( c \notin \{ 0, - 1, - 2,\ldots \rbrack \), o... | \[ \begin{array}{l} {e}_{0}^{\left( n\right) } = 0 \\ \end{array} \]\n\nassociated qd table\n\n\[ {q}_{1}^{\left( n\right) } = \frac{a + n}{c + n},\;n = 0,1,2,\ldots ,\ ]\n\n(7.1.50a)\n\n\[ {q}_{m}^{\left( n\right) } = \frac{\left( {a + n + m - 1}\right) \left( {c + n + m - 2}\right) }{\left( {c + n + {2m} - 3}\right) ... | Yes |
For the series\n\n\[ \n\\Phi \\left( {1;c;z}\\right) = 1 + \\frac{1}{c}z + \\frac{1}{c\\left( {c + 1}\\right) }{z}^{2} + \\frac{1}{c\\left( {c + 1}\\right) \\left( {c + 2}\\right) }{z}^{3} + \\cdots ,\n\]\n\nwith \( c \\notin \\left\\lbrack {0, - 1, - 2,\\ldots }\\right\\rbrack \), one obtains for the elements \( {e}_{... | \[ \n{q}_{1}^{\\left( n\\right) } = \\frac{1}{c + n},\\;n = 0,1,2,\\ldots ,\n\]\n\n\[ \n{q}_{m}^{\\left( n\\right) } = \\frac{c + n + m - 2}{\\left( {c + n + {2m} - 3}\\right) \\left( {c + n + {2m} - 2}\\right) },\\;m = 2,3,4,\\ldots ;n \\geq 0,\n\]\n\n\[ \n{e}_{m}^{\\left( n\\right) } = \\frac{-m}{\\left( {c + n + {2m... | Yes |
Theorem 7.8. Let\n\n\[ L = {c}_{0} + {c}_{1}z + {c}_{2}{z}^{2} + {c}_{3}{z}^{3} + \cdots \]\n\n(7.1.55)\n\nbe the Taylor series at \( z = 0 \) of a function \( f\left( z\right) \) holomorphic at the origin and meromorphic in the disk \( {D}_{r} = \left\lbrack {z : \left| z\right| < r}\right\rbrack \) . Let the poles \(... | Proof. Both (A) and (B) are immediate consequences of (7.1.32) and the following property of Hankel determinants (see, for example, [Henrici, 1974, Theorem 7.5b]). When the hypotheses of the theorem are satisfied, there exists a constant \( {b}_{m} \neq 0 \) independent of \( n \) such that\n\n\[ {H}_{m}^{\left( n\righ... | Yes |
Theorem 7.9. Let\n\n\\[ \n1 + \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }\\left( \\frac{{a}_{n}z}{1}\\right) ,\\;{a}_{n} \\neq 0, \n\\]\n\n(7.1.63)\n\nbe a regular \\( C \\) -fraction which converges uniformly on compact subsets of a domain \\( D \\) containing the origin to a function \\( f\\left( z\\right) \\) holo... | Proof. The theorem is a direct consequence of Theorems 5.13, 7.2, 7.5 and 7.8. | Yes |
The function\n\n\[ f\left( z\right) = 1 + \sqrt{z}\arctan \sqrt{z} \]\n\n(7.2.33)\n\nhas Taylor series at \( z = 0 \) given by\n\n\[ L = 1 + z - \frac{{z}^{2}}{3} + \frac{{z}^{3}}{5} - \frac{{z}^{4}}{7} + \frac{{z}^{5}}{9} - \cdots ,\ \]\n\n(7.2.34)\n\nconvergent for \( \left| z\right| \leq 1, z \neq - 1 \) . | Applying Algorithm 7.2.1, we obtain the coefficients \( {k}_{n},{l}_{n} \) of the corresponding associated continued fraction given in Table 7.2.1. | No |
Theorem 7.17.\n\n(A) Every general T-fraction\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\mu }{c}_{k}^{ * }{z}^{k} + \mathop{\sum }\limits_{{k = - \nu }}^{0}{c}_{k}{z}^{k} + \mathop{K}\limits_{{n = 1}}^{\infty }\left( \frac{z}{{e}_{n} + {d}_{n}z}\right) ,\;{e}_{n} \neq 0, \]\n\ncorresponds at \( z = 0 \) to a uniquely dete... | Proof. (A): Let \( {A}_{n}\left( z\right) \) and \( {B}_{n}\left( z\right) \) denote the \( n \) th numerator and denominator, respectively, of (7.3.11). Then by (7.3.10) and the determinant formulas (2.1.9) we have\n\n\[ \frac{{A}_{n + 1}\left( z\right) }{{B}_{n + 1}\left( z\right) } - \frac{{A}_{n}\left( z\right) }{{... | Yes |
Theorem 7.18. (A) If for a given pair \( \left( {L,{L}^{ * }}\right) \) of \( {fL} \) s\n\n\[ L = \mathop{\sum }\limits_{{k = - \nu }}^{\infty }{c}_{k}{z}^{k}\text{ and }{L}^{ * } = \mathop{\sum }\limits_{{k = - \infty }}^{\mu }{c}_{k}^{ * }{z}^{k},\;\mu \geq 0,\;\nu \geq 0, \]\n\n(7.3.25)\n\nthere exists a general T-f... | Proof. (A): In our proof we shall use the equivalent form (7.3.9) for the general \( T \) -fraction (7.3.26). Then by (7.3.6) and (7.3.26b), we have\n\n\[ {e}_{n} \neq 0\text{ and }{d}_{n} \neq 0,\;n = 1,2,3,\ldots \]\n\n(7.3.30)\n\nIt follows from (7.3.13) and (7.3.10) that \( L{B}_{n} - {A}_{n} \) is a fLs of the for... | Yes |
A general T-fraction\n\n\\[ \n\\mathop{K}\limits_{{n = 1}}^{\infty }\\left( \\frac{{F}_{n}z}{1 + {G}_{n}z}\\right) ,\\;\\text{ with }\\;{F}_{n} \\neq 0,{G}_{n} \\neq 0,\\;n = 1,2,3,\\ldots ,\n\\]\n\n(7.3.49)\n\ncorresponds to a fLs \\( {L}^{ * } = - 1 \\) at \\( z = \\infty \\) iff\n\n\\[ \n{F}_{n} = - {G}_{n},\\;n = 1... | Proof. (A): Suppose that (7.3.50) holds. Then a simple application of Theorem 5.2 shows that (7.3.49) corresponds to \\( {L}^{ * } = - 1 \\) at \\( z = \\infty \\) . In fact, we let \\( w = 1/z,{a}_{n}\\left( w\\right) = {F}_{n}/w,{b}_{n}\\left( w\\right) = 1 - {F}_{n}/w,{L}_{0} = {L}^{ * } = - 1,{L}_{n} = - {F}_{n}/w ... | Yes |
The exponential \( {e}^{-z} \) . Here we consider the convergent series\n\n\[ L = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\left( -z\right) }^{k}}{k!} = \frac{1}{\mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{z}^{k}}{k!}} = {e}^{-z}. \] | By Theorem 7.20 the general \( T \) -fraction\n\n\[ 1 + \frac{-z}{1 + z} + \frac{-\frac{1}{2}z}{1 + \frac{1}{2}z} + \frac{-\frac{1}{3}z}{1 + \frac{1}{3}z} + \cdots \]\n\ncorresponds to \( L \) at \( z = 0 \) and to \( {L}^{ * } = 0 \) at \( z = \infty \) . Note that this is essentially the same continued fraction as gi... | No |
Theorem 7.23. Let\n\n\\[ \n\\underset{n = 1}{\\overset{\\infty }{\\mathrm{K}}}\\left( \\frac{{F}_{n}z}{1 + {G}_{n}z}\\right) ,\\;{F}_{n} \\neq 0 \n\\]\n\n(7.3.74)\n\nbe a general T-fraction with the property that for each \\( M > 0 \\) there exists an \\( {n}_{M} \\) such that\n\n\\[ \n\\left| \\frac{{F}_{n}z}{\\left( ... | Proof. Parts (A), (B) and (C) can be proved in a manner very similar to the proof of Theorem 5.14. It will suffice to note that in the present case the denominator analogous to that on the right of (5.4.26) cannot vanish identically, since \\( F\\left( 0\\right) = 0 \\) and \\( {B}_{n}\\left( 0\\right) = 1 \\) for all ... | Yes |
The error function erf \( z \), defined by (6.1.37), is represented by the equation\n\n\[ \frac{\sqrt{\pi }}{2z}{e}^{{z}^{2}}\operatorname{erf}z = \Phi \left( {1;\frac{3}{2};{z}^{2}}\right) \] | \[ = \frac{\frac{1}{2}}{\frac{1}{2} - {z}^{2}} + \frac{1 \cdot {z}^{2}}{\frac{3}{2} - {z}^{2}} + \frac{2 \cdot {z}^{2}}{\frac{5}{2} - {z}^{2}} + \frac{3 \cdot {z}^{2}}{\frac{7}{2} - {z}^{2}} + \cdots \;\text{ for all }\;z \] | Yes |
\[ {\int }_{0}^{z}{e}^{{t}^{2}}{dt} = \frac{i\sqrt{\pi }}{2}\operatorname{erf}\left( {-{iz}}\right) = z{e}^{{z}^{2}}\Phi \left( {1;\frac{3}{2}; - {z}^{2}}\right) \] | \[ = z{e}^{{z}^{2}}\left( {\frac{\frac{1}{2}}{\frac{1}{2} + {z}^{2}} - \frac{{z}^{2}}{\frac{3}{2} + {z}^{2}} - \frac{2{z}^{2}}{\frac{5}{2} + {z}^{2}} - \frac{3{z}^{2}}{\frac{7}{2} + {z}^{2}} - \cdots }\right) \] | Yes |
\[ C\left( z\right) + {iS}\left( z\right) = \frac{1 + i}{2}\operatorname{erf}\left( {\frac{\sqrt{\pi }}{2}\left( {1 - i}\right) z}\right) = z{e}^{i\left( {\pi /2}\right) {z}^{2}}\Phi \left( {1;\frac{3}{2}; - \frac{i\pi }{2}{z}^{2}}\right) \] | \[ = z{e}^{i\left( {\pi /2}\right) {z}^{2}}\left( {\frac{\frac{1}{2}}{\frac{1}{2} + \frac{i\pi }{2}{z}^{2}} - \frac{\frac{i\pi }{2}{z}^{2}}{\frac{3}{2} + \frac{i\pi }{2}{z}^{2}} - \frac{2\left( \frac{i\pi }{2}\right) {z}^{2}}{\frac{5}{2} + \frac{i\pi }{2}{z}^{2}} - \frac{3\left( \frac{i\pi }{2}\right) {z}^{2}}{\frac{7}... | Yes |
THEOREM 7.27. Let \( a \) and \( b \) be constants satisfying \( \operatorname{Re}\left( a\right) = 0 \) and \( b > 0 \) .\n\n(A) If \( {h}_{1} \) and \( {h}_{2} \) are positive rational functions, then\n\n\[ {h}_{1} + {h}_{2},\;a + {h}_{1},\;b{h}_{1}\text{ and }1/{h}_{1} \]\n\n(7.4.11)\n\nare all positive.\n\n(B) If \... | Proof. The assertions of the theorem are simple consequences of the definition of positive and para-odd. | No |
Corollary 7.33. Let\n\n\\[ f\\left( z\\right) = {z}^{n} + {a}_{n - 1}{z}^{n - 1} + {a}_{n - 2}{z}^{n - 2} + \\cdots + {a}_{1}z + {a}_{0} \\]\n\n(7.4.36)\n\nbe a polynomial with real coefficients \\( {a}_{k} \\) . Let\n\n\\[ g\\left( z\\right) = {a}_{n - 1}{z}^{n - 1} + {a}_{n - 3}{z}^{n - 3} + {a}_{n - 5}{z}^{n - 5} + ... | The division process described in the proof of Theorem 7.30(A) can be used (when possible) to obtain the continued-fraction representation of \\( t\\left( z\\right) \\) of the form (7.4.32) or (7.4.38). The process involves only the division of polynomials and is easily programmable. The procedure is illustrated by the... | No |
Determine whether the real polynomial\n\n\[ f\left( z\right) = {z}^{4} + 5{z}^{3} + {10}{z}^{2} + {10z} + 4 \]\n\nis stable. | The test fraction of Corollary 7.33 is \( t = g/f \), where\n\n\[ g\left( z\right) = 5{z}^{3} + {10z}. \]\n\nDivision of \( f \) by \( g \) yields\n\n\[ t\left( z\right) = \frac{1}{f\left( z\right) /g\left( z\right) } = \frac{1}{1 + {h}_{1}\left( z\right) }, \]\n\nwhere\n\n\[ {h}_{1}\left( z\right) = \frac{1}{5}z + \fr... | Yes |
For what real values of the constant \( c \) will the polynomial\n\n\[ f\left( z\right) = {z}^{4} + 5{z}^{3} + {10}{z}^{2} + {10z} + c \] \n\nbe stable? | Following the procedure used in Example 2, we obtain for the test function\n\n\[ t\left( z\right) = \frac{1}{1 + \frac{1}{5}z} + \frac{1}{\frac{5}{8}z} + \frac{1}{{d}_{3}z} + \frac{1}{{d}_{4}z}, \]\n\nwhere\n\n\[ {d}_{3} = \frac{8}{{10} - \frac{5}{8}c}\;\text{ and }\;{d}_{4} = \frac{{10} - \frac{5}{8}c}{c}. \]\n\nThere... | Yes |
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