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Proposition 1.5. The are infinitely many primes. | Proof. Suppose that a complete list of primes is \( \left\{ {{p}_{1},\ldots ,{p}_{k}}\right\} \) . Consider the number \( N \mathrel{\text{:=}} {p}_{1}{p}_{2}\cdots {p}_{k} + 1 \) . This must certainly have a prime factor. However, it is easy to see that none of the primes \( {p}_{1},\ldots ,{p}_{k} \) divides \( N \),... | Yes |
Lemma 3.1. We have \( \# {\left( \mathbb{Z}/q\mathbb{Z}\right) }^{ \times } = \phi \left( q\right) \) . | Proof. Every element of \( \mathbb{Z}/q\mathbb{Z} \) is congruent to precisely one element of \( \{ 1,\ldots, q\} \) , and the elements of \( {\left( \mathbb{Z}/q\mathbb{Z}\right) }^{ \times } \) correspond to the elements of this set which are coprime to \( q \) . | Yes |
Proposition 3.1. Suppose that \( {q}_{1},\ldots ,{q}_{k} \) are pairwise coprime positive integers. Then\n\n\[ \n{\left( \mathbb{Z}/{q}_{1}\mathbb{Z}\right) }^{ \times } \times \cdots {\left( \mathbb{Z}/{q}_{k}\mathbb{Z}\right) }^{ \times } \cong {\left( \mathbb{Z}/{q}_{1}\ldots {q}_{k}\mathbb{Z}\right) }^{ \times } \n... | Proof. The proof of the Chinese remainder theorem adapts almost immediately to give this. Consider once again the map\n\n\[ \n\psi : \mathbb{Z}/{q}_{1}\cdots {q}_{k}\mathbb{Z} \rightarrow \mathbb{Z}/{q}_{1}\mathbb{Z} \times \cdots \times \mathbb{Z}/{q}_{k}\mathbb{Z} \n\]\n\ngiven by\n\n\[ \n\psi \left( {x + {q}_{1}\cdo... | Yes |
Proposition 3.3. Let \( f\left( X\right) \in R\left\lbrack X\right\rbrack \) be a polynomial of degree \( d \geq 0 \) over an integral domain \( R \) . Then \( f \) has at most \( d \) roots in \( R \) . | Proof. If \( f \) has no roots we are done, so let’s suppose \( \alpha \in R \) is a root.\n\nBy the division algorithm for polynomials \( f\left( X\right) = \left( {X - \alpha }\right) q\left( X\right) + c \) for some \( c \in R \) . (See Prelims for polynomials over the reals; the same works for a general integral do... | Yes |
Proposition 3.4. \( {\left( \mathbb{Z}/q\mathbb{Z}\right) }^{ \times } \) is cyclic if and only if \( q \) is \( 2,4 \), an odd prime power or twice an odd prime power. | Proof. We use two facts from group theory: first, that a product \( G = {C}_{{n}_{1}} \times \cdots \times \) \( {C}_{{n}_{r}} \) is not cyclic unless \( {n}_{1},\ldots ,{n}_{r} \) are coprime. (In fact this is an if and only if, but the direction we need is rather easy: every element in this group has order dividing t... | Yes |
Proposition 4.1 (Euler’s criterion). Let \( p \) be an odd prime, and suppose that \( a \in {\left( \mathbb{Z}/p\mathbb{Z}\right) }^{ \times } \) . Then\n\n\[ \n{a}^{\left( {p - 1}\right) /2} \equiv \left\{ \begin{array}{ll} 1\left( {\;\operatorname{mod}\;p}\right) & \text{ if }a\text{ is a quadratic residue }\left( {\... | Proof. By Fermat’s Little Theorem, \( {\left( {a}^{\left( {p - 1}\right) /2}\right) }^{2} \equiv 1\left( {\;\operatorname{mod}\;p}\right) \), and hence \( {a}^{\left( {p - 1}\right) /2} \equiv \) \( \pm 1\left( {\;\operatorname{mod}\;p}\right) \) . Since an equation of degree \( d < p \) has at most \( d \) roots, ther... | Yes |
Proposition 4.3 (Gauss’s Lemma). Let \( p \) be an odd prime. Let \( I \subset {\left( \mathbb{Z}/p\mathbb{Z}\right) }^{ \times } \) be a set such that \( {\left( \mathbb{Z}/p\mathbb{Z}\right) }^{ \times } \) is the disjoint union of \( I \) and \( - I = \{ - i : i \in I\} \) . Let a be an integer coprime to \( p \) . ... | Proof. Write\n\n\[ \n{J}_{ - } \mathrel{\text{:=}} \{ j \in I : {aj} \in - I\} \n\] \n\nand \n\n\[ \n{J}_{ + } \mathrel{\text{:=}} \{ j \in I : {aj} \in I\} . \n\] \n\nThen \( {J}_{ - },{J}_{ + } \) are disjoint and their union is \( I \) .\n\nWe claim that \( - a{J}_{ - } \) and \( a{J}_{ + } \) (where \( {aX} \mathre... | Yes |
Proposition 5.1. Suppose that \( n \equiv 3\left( {\;\operatorname{mod}\;4}\right) \) . Then \( n \) is prime if and only if the following is true: for every \( a \in {\left( \mathbb{Z}/n\mathbb{Z}\right) }^{ \times },{a}^{\left( {n - 1}\right) /2} \equiv \pm 1\left( {\;\operatorname{mod}\;n}\right) \) . | Proof. We have already shown the \ | No |
Proposition 5.2 (Pepin’s test). Suppose that \( n = {F}_{k}, k \geq 1 \) . Then \( n \) is prime if and only if \( {3}^{\left( {n - 1}\right) /2} \equiv - 1\left( {\;\operatorname{mod}\;n}\right) \) . | Proof. Suppose first that the congruence holds. Then certainly \( {3}^{n - 1} \equiv 1\left( {\;\operatorname{mod}\;n}\right) \) . However, as we remarked earlier, conditions like this do not imply that \( n \) is prime\n\n--- \n\n\( {}^{1} \) The statement that all nontrivial zeros of all Dirichlet \( L \) -functions ... | No |
Property \( B \) If the vectors \( {v}_{1},{v}_{2},\ldots ,{v}_{n} \) are linearly dependant, then\n\n\[ D\left( {{v}_{1},{v}_{2},\ldots ,{v}_{n}}\right) = 0. \] | Proof By the definition of linearly independent vectors, at least one of the vectors can be written as a linear combination of the others. Without loss of generality suppose we can write \( {v}_{1} = {c}_{2}{v}_{2} + \ldots + {c}_{n}{v}_{n} \), where the \( {c}_{i} \) are scalars. By repeated uses of Property 3 we have... | Yes |
Theorem 2.1 If \( f : {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) is a differentiable function of \( x \) and \( y \), then \( f \) has directional derivatives in the direction of any unit vector \( u = \left\lbrack \begin{array}{l} a \\ b \end{array}\right\rbrack \) and\n\n\[ \n{D}_{u}f\left( {x, y}\right) = \frac{\par... | Sketch of proof: First define a one variable function \( g\left( h\right) \) by \( g\left( h\right) = f\left( {{x}_{0} + {ha},{y}_{0} + {hb}}\right) \) and use this to rewrite \( {D}_{u}f \) . Then use the chain rule to write the derivative of \( g\left( h\right) \) . Finally, let \( h = 0 \) and combine. | No |
Theorem 11.1 (Stokes’ Theorem) Let \( M \) be a smooth oriented \( n \) -dimensional manifold and let \( \alpha \) be an \( \left( {n - 1}\right) \) -form on M. Then\n\n\[{\int }_{M}{d\alpha } = {\int }_{\partial M}\alpha\]\n\nwhere \( \partial M \) is given the induced orientation. | Its proof appears in pretty much all the standard texts of differential geometry and is done with various levels of abstraction and rigor. As with much of this book we will try to strike a balance between understandability, rigor, and abstraction. In particular, the proof we give works for \ | No |
Problem 2. Find the mean square fluctuation of the velocity. | Solution. The result of Problem 1 with \( n = 1 \) and \( n = 2 \) gives\n\n\[ \left\langle {\left( \Delta v\right) }^{2}\right\rangle = \mathrm{f} - \mathrm{{es}} = \left( {\mathrm{T}/\mathrm{m}}\right) \left( {3 - 8/\mathrm{n}}\right) .\n\] | Yes |
Problem 3. Find the mean energy, the mean square energy, and the mean square fluctuation of the kinetic energy of an atom. | Solution. From the results of Problem 1 we find\n\n\[ \bar{\varepsilon } = \frac{1}{2}m\overline{{v}^{2}} = {3T}/2 \]\n\n\[ \overline{{\varepsilon }^{2}} = {15}{T}^{2}/4 \]\n\n\[ \left\langle {\left( \Delta \varepsilon \right) }^{2}\right\rangle = 3{T}^{2}/2 \] | Yes |
Problem 4. Find the probability distribution for the kinetic energy of an atom. | \[ \mathrm{d}{w}_{\varepsilon } = \frac{2}{\sqrt{\left( \pi {T}^{3}\right) }}{e}^{-\varepsilon /T}\sqrt{}\varepsilon \mathrm{d}\varepsilon . \] | Yes |
Problem 5. Find the probability distribution for the angular velocities of rotation of molecules. | Solution. Just as for translational motion, we can write the probability distribution for the rotation of each molecule separately (in classical statistics). The kinetic energy of rotation of a molecule regarded as a rigid body (which is permissible, owing to the smallness of the atomic vibrations within the molecule) ... | Yes |
Problem 1. The potential energy of the interaction between the particles in a body is a homogeneous function of degree \( n \) in their coordinates. Using similarity arguments, determine the form of the free energy of such a body in classical statistics. | SOLUTION. In the partition function\n\n\[ Z = {\int }^{\prime }{e}^{-\left\lbrack {K\left( p\right) + U\left( q\right) }\right\rbrack /T}\mathrm{\;d}I, \]\n\nwe replace each \( q \) by \( {\lambda q} \) and each \( p \) by \( {\lambda }^{n/2}p \), where \( \lambda \) is an arbitrary constant. If at the same time we rep... | Yes |
Problem 2. Derive the virial theorem for a macroscopic body for which the potential energy of interaction of the particles is a homogeneous function of degree \( \mathbf{n} \) in their coordinates. | Solution. Following the derivation of the virial theorem in mechanics (see Mechanics, \( §{10} \) ). we calculate the time derivative of the sum&p. where \( \mathbf{r} \) and \( \mathbf{p} \) are the rad s vectors and momenta of the particles in the body. Since \( \dot{\mathbf{r}} = \partial \mathbf{K}\left( p\right) /... | Yes |
Problem 1. Find the density of gas in a cylinder of radius \( R \) and length \( l \) rotating about its axis with angular velocity \( \Omega \), there being a total of \( \mathbf{N} \) molecules in the cylinder. | Solution . It has \( {been} \) mentioned in § 34 that the rotation of a body as a whole is equivalent to the presence of an external field with potential energy \( - \frac{1}{2}m{\Omega }^{2}{r}^{2} \) (where \( r \) is the distance from the axis of rotation). The gas density is therefore\n\n\[ n\left( r\right) = A{e}^... | Yes |
Problem 1. Find the number of impacts of gas molecules on unit area of the wall per unit time for which the angle between the direction of the velocity of the molecule and the normal to the surface lies between \( \theta \) and \( \theta + \mathrm{d}\theta \) . | \[ \mathrm{d}{v}_{\theta } = \frac{N}{V}{\left( \frac{2T}{m\pi }\right) }^{1/2}\sin \theta \cos \theta \mathrm{d}\theta . \] | Yes |
Problem 2. Find the number of impacts of gas molecules on unit area of the wall per unit time for which the absolute magnitude of the velocity lies between \( v \) and \( v + \mathrm{d}v \) . | \[ \mathrm{d}{v}_{v} = \frac{N}{V}\pi {\left( \frac{m}{2\pi T}\right) }^{3/2}{e}^{-m{v}^{2}/{2T}}{v}^{3}\mathrm{\;d}v. \] | Yes |
Problem 3. Find the total kinetic energy \( {E}_{\text{inc }} \) of the gas molecules striking unit area of the wall per unit time. | \[ \n{E}_{\text{inc }} = \frac{N}{V}\sqrt{\frac{2{T}^{3}}{m\pi }} = P\sqrt{\frac{2T}{m\pi }}. \n\] | Yes |
Problem 4. Find the number of collisions between one molecule and the rest per unit time, assuming the molecules to be rigid spheres of radius \( r \) . | Solution. The cross-section for collisions between molecules is then \( \sigma = \) \( \pi {\left( 2r\right) }^{2} = {4\pi }{r}^{2} \) (since a collision occurs whenever two molecules pass at a distance less than \( {2r} \) ). Substitution in (39.5) gives\n\n\[ r = {16}{r}^{2}\sqrt{\frac{\pi T}{m}}\frac{N}{V} = {16}{r}... | No |
Problem 1. Find the work done on an ideal gas in an isothermal change of volume from \( {\mathbf{V}}_{\mathbf{1}} \) to \( {\mathbf{V}}_{\mathbf{2}} \) (or of pressure from \( {\mathbf{P}}_{\mathbf{1}} \) to \( {\mathbf{P}}_{\mathbf{2}} \) ). | Solution. The required work \( R \) is equal to the change in the free energy of the gas. and from (42.4) we have\n\n\[ \mathbf{R} = {\mathbf{F}}_{2} - {\mathbf{F}}_{1} = \mathbf{{NT}}\log \left( {{V}_{1}/{V}_{2}}\right) = \mathbf{{NT}}\log \left( {{\mathbf{P}}_{2}/{\mathbf{P}}_{1}}\right) . \]\n\nThe quantity of heat ... | Yes |
Problem 2. Two vessels contain two identical ideal gases at the same temperature \( \mathbf{T} \) and with equal numbers of particles \( \mathrm{N} \) but at different pressures \( {\mathbf{P}}_{\mathbf{1}} \) and \( {\mathbf{P}}_{\mathbf{2}} \) . The vessels are then connected. Find the change in entropy. | Solution. Before the vessels are connected, the entropy of the two gases is equal to the sum of their entropies. \( {S}_{0} = - \mathrm{N}\log \left( {{P}_{1}{P}_{2}}\right) - {2N}{\chi }^{\prime }\left( T\right) \) . After the connection, the temperature of the gases remains the same (as follows from the conservation ... | Yes |
Problem 3. Find the energy of an ideal gas in a cylindrical vessel of radius \( \\mathbf{R} \) and length \( \\mathbf{l} \) rotating about its axis with angular velocity \( \\mathbf{\\Omega } \) . | SOLUTION. According to § 34. the rotation is equivalent to the presence of an external lentrifugal \( f \) ield with potential energy \( u = - \\frac{1}{2}m{\\Omega }^{2}{r}^{2}(r \) being the distance of a particle from the axis of rotation). \n\nWhen an external field is present, the integrand in (42.2) contains an e... | Yes |
Problem 1. Two identical ideal gases at the same pressure \( \mathrm{P} \) and containing the same number of particles \( \mathrm{N} \) but at different temperatures \( {\mathbf{T}}_{\mathbf{1}} \) and \( {\mathbf{T}}_{\mathbf{2}} \) are in vessels with volumes \( {\mathbf{V}}_{\mathbf{1}} \) and Vs. The vessels are th... | Solution. Before the vessels are connected, the entropy of the two gases, equal to the sum of their entropies, is by (43.6) \( {S}_{0} = - 2\mathrm{\;N}\log P + N{c}_{p}\log \left( {{T}_{1}{T}_{2}}\right) {.}^{ \dagger } \) After the connection, the temperatures of the gases become equal. The sum of the energies of the... | Yes |
Problem 2. Find the work done on an ideal gas in adiabatic compression. | Solution. In an adiabatic process the quantity of heat \( \mathrm{Q} = 0 \), and so \( R = \) \( {\mathbf{E}}_{\mathbf{2}} - {\mathbf{E}}_{\mathbf{1}} \), where \( {\mathbf{E}}_{\mathbf{2}} - {\mathbf{E}}_{\mathbf{1}} \) is the change in energy during the process. According to (43.2) \( R = N{c}_{v}\left( {{T}_{2} - {T... | Yes |
Problem 4. Find the work done and quantity of heat gained in an isobaric process, i.e. one which occurs at constant pressure. | Solution. At constant pressure\n\n\[ \nR = - P\left( {{V}_{2} - {V}_{1}}\right) ,\;Q = {W}_{2} - {W}_{1}, \]\n\nwhence\n\n\[ \nR = N\left( {{T}_{1} - {T}_{2}}\right) ,\;Q = N{c}_{p}\left( {{T}_{2} - {T}_{1}}\right) . \]\n | No |
Problem 5. Find the work done on a gas and the quantity of heat which it gains in compression from volume \( {V}_{1} \) to \( {V}_{2} \) in accordance with the equation \( \mathbf{P}{V}^{n} = a \) (a polytropic process). | SoLution. The work is\n\n\[ R = - {\int }_{{V}_{1}}^{{V}_{2}}{PdV} = \frac{a}{n - 1}\left( {{V}_{2}^{1 - n} - {V}_{1}^{1 - n}}\right) .\n\]\n\nSince the sum of the quantity of heat gained and the work done is equal to the total change in energy, we have \( \mathrm{Q} = N{c}_{v}\left( {{T}_{2} - {T}_{1}}\right) - R \), ... | Yes |
Problem 6. Find the work done on an ideal gas and the quantity of heat which it gains on going through a cyclic process (i.e. one in which it returns to its initial state at the end of the process). consisting of two isochoric and two isobaric processes: the gas goes from a state with pressure and volume \( {\mathbf{P}... | Solution. The change in energy in a cyclic process is zero, since the initial and final states are the same. The work done and the quantity of heat gained in such a process are therefore the same with opposite signs \( \left( {R = - Q}\right) \) . In order to find \( R \) in the present case, we note that in isochoric ... | Yes |
Problem 7. The same as Problem 6, but for a cyclic process consisting of two isochoric and two isothermal processes. the successive volumes and temperatures of the gas being \( {V}_{1},{T}_{1};{V}_{1},{T}_{2};{V}_{2},{T}_{2};{V}_{2},{T}_{1};{V}_{1},{T}_{1} \) . | \[ R = \left( {{T}_{2} - {T}_{1}}\right) N\log \left( {{V}_{1}/{V}_{2}}\right) \] | Yes |
Problem 8. The same as Problem 6. but for a cyclic process consisting of two isothermal and two adiabatic processes, the successive entropies. temperatures and pressures being \( \mathrm{S},{T}_{1},{P}_{1};{S}_{1},{T}_{2};{S}_{2},{T}_{2},{P}_{2};{S}_{2},{T}_{1};{S}_{1},{T}_{1},{P}_{1} \) . | SOLUTION.\n\n\[ Q = \left( {{T}_{2} - {T}_{1}}\right) \left( {{S}_{2} - {S}_{1}}\right) \]\n\n\[ = \left( {{T}_{2} - {T}_{1}}\right) \left\lbrack {N\log \left( {{P}_{1}/{P}_{2}}\right) + N{c}_{p}\log \left( {{T}_{2}/{T}_{1}}\right) }\right\rbrack . \] | Yes |
Problem 9. The same as Problem 6, but for a cyclic process consisting of two isobaric and two isothermal processes, the successive states being-P,, \( {T}_{1};{P}_{1},{T}_{2} \) ; \( {P}_{2},{T}_{2};{P}_{2},{T}_{1} : {P}_{1},{T}_{1} \) . | solution. The work done on the gas in the isobaric processes is (see Problem 4) \( N\left( {{T}_{1} - {T}_{2}}\right) \) and \( N\left( {{T}_{2} - {T}_{1}}\right) \), and that in the isothermal processes is \( {NT},\log \left( {{P}_{2}/{P}_{1}}\right) \) and \( {NT},\log \left( {{P}_{1}/{P}_{2}}\right) \). The sum of t... | Yes |
Problem 10. The same as Problem 6, but for a cyclic process consisting of two isobaric and two adiabatic processes. the successive states being \( {P}_{1}, S,,{T}_{1};{P}_{1} \) , \( {S}_{2};{P}_{2},{S}_{2},{T}_{2};{P}_{2}, S,;{P}_{1},{S}_{1},{T}_{1} \) . | solution. The temperature in the second state is \( {T}_{2}{\left( {P}_{2}/{P}_{1}\right) }^{\left( {1 - \gamma }\right) /\gamma } \), and in the fourth state \( {\mathbf{T}}_{1}{\left( {\mathbf{P}}_{1}/{\mathbf{P}}_{2}\right) }^{\left( {1 - \gamma }\right) /\gamma } \) ; these are obtained from \( {\mathbf{T}}_{1} \) ... | Yes |
Problem 11. The same as Problem 6. but for a cyclic process consisting of two isochoric and two adiabatic processes, the successive states being \( {\mathbf{V}}_{1},\mathrm{\;S},,{\mathbf{T}}_{1} \) ; \( {V}_{1},{S}_{2};{V}_{2},{S}_{2},{T}_{2};{V}_{2},{S}_{1};{V}_{1},{S}_{1},{T}_{1} \) . | Solution. Using the result of Problem 2, we find\n\n\[ \nR = N{c}_{v}{T}_{2}\left\lbrack {1 - {\left( {V}_{2}/{V}_{1}\right) }^{\gamma - 1}}\right\rbrack + N{c}_{v}{T}_{1}\left\lbrack {1 - {\left( {V}_{1}/{V}_{2}\right) }^{\gamma - 1}}\right\rbrack .\n\] | Yes |
Problem 12. Determine the maximum work that can be obtained by connecting vessels containing two identical ideal gases at the same temperature \( {\mathbf{T}}_{\mathbf{0}} \) and with equal numbers of particles \( \mathrm{N} \) but having different volumes \( {\mathbf{V}}_{\mathbf{1}} \) and \( {\mathbf{V}}_{\mathbf{2}... | solution. The maximum work is done if the process occurs reversibly (i.e. if the entropy remains constant), and is equal to the difference between the energies before and after the process (§ 19). Before the connection of the vessels, the entropy of the two gases is equal to the sum of their entropies. i.e. by (43.5)\n... | Yes |
Problem 13. The same as Problem 12, but for gases with the same pressure \( {P}_{0} \) and different temperatures \( {T}_{1} \) and \( {T}_{2} \) before the connection of the vessels. | Solution. We have similarly\n\n\[ \n{R}_{\max } = N{c}_{v}\left\{ {{T}_{1} + {T}_{2} - {2}^{\gamma }\sqrt{}\left( {{T}_{1}{T}_{2}}\right) {\left\lbrack \frac{{T}_{1}{T}_{2}}{{\left( {T}_{1} + {T}_{2}\right) }^{2}}\right\rbrack }^{\left( {\gamma - 1}\right) /2}}\right\} .\n\] | Yes |
Problem 14. Find the minimum work that must be done on an ideal gas in order to compress it from pressure \( {P}_{1} \) to \( {P}_{2} \) at a constant temperature equal to that of the surrounding medium \( \left( {T = {T}_{0}}\right) \) . | Solution. According to (20.2) the minimum work is \( {\mathbf{R}}_{\min } \) - (E--E,)- \n\n- \( {T}_{0}\left( {{S}_{2} - {S}_{1}}\right) + {P}_{0}\left( {{V}_{2} - {V}_{1}}\right) \), where the suffixes 1 and 2 refer to the gas before and after compression. In the present case the energy \( \mathbf{E} \) is unchanged ... | Yes |
Problem 15. Determine the maximum work which can be obtained from an ideal gas cooled from temperature \( \mathbf{T} \) to the temperature of the medium \( {\mathbf{T}}_{0} \) at constant volume. | Solution. From the general formula (20.3), \[ {R}_{\max } = N{c}_{v}\left( {T - {T}_{0}}\right) + N{c}_{v}{T}_{0}\log \left( {{T}_{0}/T}\right) . \] | Yes |
Problem 16. The same as Problem 15, but for a gas cooled from temperature \( T \) to the temperature of the medium \( {T}_{0} \) and at the same time expanding from pressure \( \mathbf{P} \) to the pressure of the medium \( {\mathbf{P}}_{\mathbf{0}} \) . | \[ {R}_{\max } = N{c}_{v}\left( {T - {T}_{0}}\right) + {NT},\;\log \left( {P/P,}\right) + N{c}_{p}{T}_{0}\log \left( {{T}_{0}/T}\right) + N\left( {T{P}_{0}/P - {T}_{0}}\right) . \] | Yes |
Problem 17. Gas at temperature \( {T}_{0} \) flows from a large thermally isolated reservoir into an empty thermally isolated vessel, the gas pressure in the reservoir remaining constant. Find the change in the gas temperature. | Solution. The energy \( E \) of the gas in the vessel consists of the energy \( {E}_{0} \) which it had in the reservoir and the work done on it to \ | No |
Problem 2. Determine the magnetic susceptibility of a diatomic gas when the tine-structure spacings of the electron ground state of the molecule are large in comparison with \( T.{}^{ \dagger } \) | Solution. In this case it is sufficient to consider only the ground level of the molecule, i.e. the lowest component of the ground-state multiplet. The mean value of the magnetic moment of the molecule in a state where the projections of the orbital angular momentum and the spin on the axis of the molecule are \( \Lamb... | Yes |
Problem 3. The same as Problem 2, but with the fine-structure spacings small in comparison with \( \mathbf{T} \) (molecular term case b). | Solution. In this case the averaging must be taken over all components of the multiplet. The diagonal matrix elements of the z-component of the magnetic moment for given values of \( \Lambda \) and the spin z-component \( {\mathbf{M}}_{\mathbf{S}} \) are\n\n\[ \left\langle {\Lambda {M}_{S}\left| {\mathfrak{m}}_{z}\righ... | Yes |
Problem 4. Determine the magnetic susceptibility of the gas NO. The electron ground term of the molecule is \( {}^{2}\Pi \) (i.e. \( \Lambda = 1, S = \frac{1}{2} \) ), and the spacing \( \Delta \) between the doublet components is comparable \( {}^{ \dagger } \) to the temperature \( \mathbf{T} \) (J. H. Van Vleck, 192... | Solution. Here, in the averaging in (52.6). we have to take account of both components of the doublet level with different Boltzmann factors. The diagonal matrix elements of the magnetic moment for the two states \( |{A\sum }\rangle \) are\n\n\[ \text{(1.} - \frac{1}{2}\left| {\mathbf{L} + 2\mathbf{S}}\right| \mathbf{i... | Yes |
Problem 2. Determine the specific heat of a degenerate extreme relativistic electron gas. | Solution. Applying the formula (58.1) to the integral in (61.6). we find\n\n\[ \Omega = {\Omega }_{0} - \frac{{\left( \mu T\right) }^{2}}{6{\left( c\hslash \right) }^{3}}V \]\n\nHence the entropy\n\[ S = \frac{{\mu }^{2}}{3{\left( c\hslash \right) }^{3}}{VT} = N\frac{{\left( 3{\pi }^{2}\right) }^{2/3}}{{3c}\hslash }T{0... | No |
Problem 3. Determine the equation of state of a relativistic completely degenerate electron gas (the electron energy and momentum being related by \( {\varepsilon }^{2} = \\left. {{c}^{3}{p}^{3} + {m}^{2}{c}^{4}}\\right) \) . | Solution. The previous formulae (57.1) and (57.2) give the number of states and the limiting momentum. and the total energy is\n\n\[ E = \\frac{Vc}{{\pi }^{2}{\\hslash }^{3}}{\\int }_{0}^{{p}_{F}}{p}^{2}\\sqrt{}\\left( {{m}^{2}{c}^{2} + {p}^{2}}\\right) \\mathrm{d}p \]\n\nwhence\n\n\[ E = \\frac{cV}{8{\\pi }^{2}{\\hsla... | Yes |
Problem 1. Determine the maximum work which can be obtained from two identical solid bodies at temperatures \( {T}_{1} \) and \( {T}_{2} \) when their temperatures are made equal. | Solution. The solution is similar to that in \( §{43} \), Problem 12, and gives\n\n\[ \n{\left| R\right| }_{\max } = {Nc}{\left( \sqrt{}{T}_{1} - \sqrt{}{T}_{2}\right) }^{2}.\n\] | Yes |
Problem 2. Determine the maximum work which can be obtained from a solid when it is cooled from a temperature \( \mathbf{T} \) to the temperature \( {\mathbf{T}}_{0} \) of the medium (at constant volume). | Solution. From formula (20.3) we have\n\n\[ \n{\left| R\right| }_{\max } = {Nc}\left( {T - {T}_{0}}\right) + {Nc}{T}_{0}\log \left( {{T}_{0}/T}\right) .\n\] | Yes |
Problem 1. Determine \( B\\left( T\\right) \) for a gas whose particles repel one another according to \( {U}_{12} = \\alpha /{r}^{n}\\left( {n > 3}\\right) \) . | Solution. In (74.5) we put \( \\mathbf{d}\\mathbf{V} = \\mathbf{4}\\pi {\\mathbf{r}}^{2} \) dr and integrate by parts with respect to \( r \) from 0 to \( \\infty \) ; the substitution \( \\alpha /T{r}^{n} = \\mathbf{x} \) then reduces the integral to a gamma function:\n\n\[ B\\left( T\\right) = \\frac{2\\pi }{3}{\\lef... | Yes |
Problem 2. The figacity of a gas is the pressure \( {\mathbf{P}}^{ \bullet } \) which it would have for given values of the temperature and chemical potential if so rarefied that it could be regarded as an ideal gas. Determine the fugacity of a gas with the thermodynamic potential (74.7). | Solution. The chemical potential of the gas is (with \( {\mu }_{\mathrm{{id}}} \) given by (42.6))\n\n\[ \mu = {\mu }_{\mathrm{{id}}} + {BP} = T\log P + \chi \left( T\right) + {BP}. \]\n\nEquating this to \( \mathbf{T}\log {P}^{ * } + \chi \left( T\right) \) by the definition of the fugacity. we have to the same accura... | Yes |
Problem 1. Find \( {C}_{p} - {C}_{v} \) for a non-ideal gas described by van der Waalsí formula. | Solution. Using formula (16.10) and van der Waalsí equation. we find\n\n\[ \n{C}_{p} - {C}_{v} = \frac{\mathrm{N}}{1 - 2\mathrm{\;N}a{\left( V - \mathrm{N}b\right) }^{2}/T{V}^{3}}. \n\] | Yes |
Problem 2. Find the equation of an adiabatic process for a van der Waals gas of constant specific heat \( \mathrm{C} \) . | Solution. Substituting in (76.8) \( {S}_{\mathrm{{id}}} = N\log V + N{c}_{v}\log T \) (omitting unimportant constants) and putting \( S = \) constant, we obtain the relation \( \left( {V - {Nb}}\right) {T}^{{c}_{v}} = \) constant. This differs from the corresponding equation for an ideal gas in that \( V \) is replaced... | No |
Problem 3. For a gas of the same kind as in Problem 2, find the change in temperature on expansion into a vacuum from volume \( {\mathbf{V}}_{\mathbf{1}} \) to \( {\mathbf{V}}_{\mathbf{2}} \) . | Solution. In an expansion into a vacuum, the energy of the gas remains con. stant. Thus formula (76.9). with \( {\mathbf{E}}_{\mathbf{{id}}} = \mathbf{N}{\mathbf{C}}_{\mathbf{v}}\mathbf{T} \), gives\n\n\[ \n{T}_{2} - {T}_{1} = \frac{Na}{{C}_{v}}\left( {\frac{1}{{V}_{2}} - \frac{1}{{V}_{1}}}\right) \n\] | Yes |
For a van der Waals gas find the temperature dependence of the inversion point for the Joule-Thomson effect. | The inversion point is determined by the equation \( {\left( \partial T/\partial V\right) }_{P} = T/V \) (see (74.9)). Substitution of \( T \) from (76.7) leads to an equation which has to be solved simultaneously with (76.7): Algebraic calculation gives the following dependence of the inversion point on pressure:\n\n\... | Yes |
Problem 1. Determine the temperature dependence of the saturated vapour pressure above a solid. The vapour is regarded as an ideal gas, and both the gas and the solid have constant specific heats. | Solution. The chemical potential of the vapour is given by formula (43.3) and that of the solid by (65.6); since the saturated vapour pressure is relatively small, the quantity \( {PV} \) may be neglected for the solid, and taken as equal to \( \mathbf{F} \). Equating the two expressions, we find \[ P = \text{ constant... | Yes |
Problem 2. Determine the rate of evaporation from a condensed state into a vacuum. | Solution. The rate of evaporation into a vacuum is determined by the number of particles which leave unit surface area of the body per unit time. Let us consider a body in equilibrium with its saturated vapour. Then the number of particles leaving the surface is equal to the number which strike and ladhere tol this sur... | Yes |
Problem 1. Determine the specific heat of a vapour along the equilibrium curve of the liquid and its saturated vapour (i.e. the specific heat for a process in which the liquid is always in equilibrium with its saturated vapour). The vapour is \( \mathbf{{re}} \) - garded as an ideal gas. | Solution. The required specific heat \( h = T\mathrm{\;d}s/\mathrm{d}T \), where \( \mathrm{d}s/\mathrm{d}T \) is the derivative along the equilibrium curve:\n\n\[ h = T\frac{\mathrm{d}s}{\mathrm{\;d}T} = T{\left( \frac{\partial s}{\partial T}\right) }_{P} + T{\left( \frac{\partial s}{\partial P}\right) }_{T}\frac{\mat... | Yes |
Problem 2. Determine the change in the volume of a vapour with temperature in a process where the vapour is always in equilibrium with the liquid (i.e. along the equilibrium curve of the liquid and its vapour). | Solution. We have to determine the derivative \( \mathrm{d}v/\mathrm{d}T \) along the equilibrium curve:\n\n\[ \frac{\mathrm{d}v}{\mathrm{\;d}T} = {\left( \frac{\partial v}{\partial T}\right) }_{P} + {\left( \frac{\partial v}{\partial P}\right) }_{T}\frac{\mathrm{d}P}{\mathrm{\;d}T}. \]\n\nSubstituting from (82.3). and... | Yes |
Problem 1. Find the variation of concentration with height for a solution in a gravitational field. | Solution. We apply the equilibrium condition (85.3) in an external field, writing it for the solute: \( T\log c + \psi \left( {P, T}\right) + {mgz} = \) constant, since the potential energy of a solute molecule in the gravitational field is \( \mathrm{{mgz}} \) (z being the height, and \( m \) the mass of the molecule)... | Yes |
Problem 2. Find the relation between the changes in the solubilities of two substances simultaneously dissolved in the same solvent. \( {}^{ \dagger } \) | Solution. The interaction between the two solutes is taken into account by the quadratic term (proportional to \( {n}_{1}{n}_{2} \) ) in the thermodynamic potential (87.3). The chemical potentials of the solutes are\n\n\[ \n{\mu }_{1}^{\prime } = \partial \Phi /\partial {n}_{1} = T\log {c}_{1} + {\psi }_{1} + {c}_{1}{\... | Yes |
Problem 3. Find the relation between the changes of saturated vapour pressure of two solutes when both are present. | Solution. The saturated vapour pressures above solutions of each substance separately are given by the equilibrium conditions\n\n\[ \n\\mathrm{T}\\log {P}_{1} + {\\chi }_{1}\\left( T\\right) = \\mathrm{T}\\log {c}_{1} + {\\psi }_{1} + {c}_{1}{\\beta }_{11}.\n\]\n\n\[ \n\\mathrm{T}\\log {P}_{2} + {\\chi }_{2}\\left( T\\... | Yes |
Problem 1. Find the maximum work that can be done in the formation of a saturated solution. | Solution. Before dissolution. the thermodynamic potential of the pure solvent was \( N{\mu }_{0} \), and that of the pure solute \( n{\mu }_{0}^{\prime } \) . The potential of the whole system was \( {\Phi }_{1} = N{\mu }_{0} + n{\mu }_{0}^{\prime } \) . After dissolution, the thermodynamic potential \( {\dot{\Phi }}_{... | Yes |
Problem 2. Find the minimum work which must be done to raise the concentration of a solution from \( {c}_{1} \) to \( {c}_{2} \) by removing some of the solvent. | Solution. Before the removal, the thermodynamic potential of the solution was \( {\Phi }_{1} = N{\mu }_{0} + N{c}_{1}T\log \left( {{c}_{1}/e}\right) + N{c}_{1}\psi \) (the number of solute molecules was \( N{c}_{1} \) , where \( \mathrm{N} \) was the original number of solvent molecules). In order to raise the concentr... | Yes |
Problem 1. Find the degree of dissociation of a diatomic gas at high temperatures, if the gas molecule consists of identical atoms and has no spin or orbital angular momentum in the ground state. | Solution. The reaction has the form \( \mathrm{A} = 2\mathrm{\;A} \) . In these Problems, we shall use the suffixes 1 and 2 to denote quantities relating respectively to the atomic (A) and molecular \( \left( {A}_{2}\right) \) components of the mixture. The degree of dissociation is defined as the ratio \( a = {N}_{1}/... | Yes |
Problem 2 The same as Problem 1, but to find the specific heat. | Solution. The entropy of the gas may be calculated as the sum\n\n\[ S = {N}_{1}\left( {{c}_{p1} + \frac{{\varepsilon }_{01} - {\mu }_{1}}{T}}\right) + {N}_{2}\left( {{c}_{p2} + \frac{{\varepsilon }_{02} - {\mu }_{2}}{T}}\right) \]\n\n\[ = {N}_{1}\left( {{c}_{p1} + \frac{{\varepsilon }_{01}}{T}}\right) + {N}_{2}\left( {... | Yes |
Problem 3. Determine the dependence of the concentration of hydrogen dissolved as \( \mathrm{H} \) atoms in a metal on the pressure of \( {\mathrm{H}}_{2} \) gas over the metal. | Solution. Regarding the process as a chemical reaction \( {\mathrm{H}}_{2} = 2\mathrm{H} \), we can write the equilibrium condition as \( {\mu }_{{\mathbf{H}}_{\mathbf{z}}} = 2{\mu }_{\mathbf{H}};{\mu }_{{\mathbf{H}}_{\mathbf{z}}} \) is written as the chemical potential of an ideal gas, \( {\mu }_{{\mathrm{H}}_{2}} = \... | Yes |
Problem 1. Find the mean square fluctuation of the energy (using \( V \) and \( T \) as independent variables). | Solution. We have\n\n\[ \n{\Delta E} = {\left( \frac{\partial E}{\partial V}\right) }_{T}{\Delta V} + {\left( \frac{\partial E}{\partial T}\right) }_{V}{\Delta T} = \left\lbrack {T{\left( \frac{\partial P}{\partial T}\right) }_{V} - P}\right\rbrack {\Delta V} + {C}_{0}{\Delta T}.\n\]\n\nSquaring and averaging, we obtai... | Yes |
Problem 3. Find \( \langle \Delta \mathbf{T}\Delta \mathbf{P}\rangle \) (with variables \( \mathbf{V} \) and \( \mathbf{T} \) ). | SOLUTION. \( \langle {\Delta T\Delta P}\rangle = \left( {{T}^{2}/{C}_{v}}\right) {\left( \partial P/\partial T\right) }_{V} \) . | Yes |
Problem 4. Find (AVAP) (with variables \( \mathbf{V} \) and \( \mathbf{T} \) ). | Solution. \( \langle {\Delta V\Delta P}\rangle = - T \) . | No |
Problem 5. Find \( \langle {\Delta S\Delta V}\rangle \) (with variables \( V \) and \( \mathrm{T} \) ). | Solution. \( \langle {\Delta S\Delta V}\rangle = {\left( \partial V/\partial T\right) }_{P}T \) . | Yes |
Problem 6. Find \( \langle {\Delta S\Delta T}\rangle \) (with variables \( \mathbf{V} \) and \( \mathbf{T} \) ). | Solution. \( \langle {\Delta S\Delta T}\rangle = T \) . | No |
Problem 9. Determine the mean value of the product of the fluctuation displacements of two different points of the string. | Solution. Let \( {y}_{1},{y}_{2} \) be the transverse movements of points at distances \( {x}_{1},{x}_{2} \) from one end of the string (with \( {x}_{2} > {x}_{1} \) ). The equilibrium form for given \( {y}_{1},{y}_{2} \) consists of three straight segments, and the work is\n\n\[ \n{R}_{\min } = \frac{1}{2}F\left( {{y}... | Yes |
Problem 1. Determine the mean square of the Fourier component (with small wave numbers: \( k \ll {p}_{F}/\hslash \) ) of the density fluctuations in a Fermi gas at \( T = 0 \) . | solution. The integrand in (117.9) is non-zero (and equal to unity) only at points where \( \overline{{n}_{\mathbf{p}}} = 1,{n}_{\mathbf{p} + \hslash \mathbf{k}} = 0 \), i.e. at points in a sphere of radius \( {\mathbf{p}}_{\mathbf{F}} \) and also not in a similar sphere whose centre is shifted by \( \hslash \mathbf{k}... | Yes |
Problem 2. Determine the correlation function for a Fermi gas at temperatures small compared with the degeneracy temperature. | Solution. In the integral in (117.8) we put \( \mu \cong {\varepsilon }_{F} = {p}_{F}^{2}/{2m} \) and transform it as follows:\n\n\[ I = {\int }_{0}^{\infty }\frac{P\sin \left( {{pr}/\hslash }\right) \mathrm{d}p}{{e}^{\left( {a - {s}_{r}}\right) /T} + 1} = - \hslash {\frac{\partial }{\partial r}}_{I}^{\infty }\frac{\co... | Yes |
Problem 3. Determine the correlation function for a Bose gas at large distances \( \\left( {r \\gg \\hslash /\\sqrt{}\\left( {mT}\\right) }\\right) \) for temperatures above the point \( {T}_{0} \) at which Bose-Einstein condensation begins, but close to this point. | Solution. Near the point \( {T}_{0} \), the chemical potential \( \\left| \\mathbf{\\mu }\\right| \) is small (see § 62, Problem). The integral in (117.7). which we denote by \( {I}_{0} \) is then determined by small values of \( p : \\varepsilon /T \\sim {p}^{2}/{mT} \\sim \\left| \\mu \\right| /T \\ll 1 \) . Hence, e... | Yes |
Problem 4. Determine the correlation function for a Bose gas with \( T < {T}_{0} \) . | Solution. For \( T < {T}_{0} \), a finite fraction of the number of particles \( \left( {N}_{0 = 0}\right) \) is in states with \( \mathbf{p} = \mathbf{0} \) (the condensate). Returning to the expression (117.4). we must first (before changing from summation to integration) separate the terms with \( \mathbf{p} \) or \... | Yes |
Problem 1. Determine the probability of formation of a nucleus of a liquid on a solid surface for a given (non-zero) value of the angle of contact \( \mathbf{\theta } \) . | Solution. The nucleus will have the shape of a segment of a sphere with base radius rsin \( \theta, r \) being the radius of the sphere. Its volume is \( V = \frac{1}{3}\pi {r}^{3}{\left( 1 - \cos \theta \right) }^{2} \) \( \left( {2 + \cos \theta }\right) \), and the surface areas of the curved part and the base are r... | Yes |
Problem 2. Find the probability of formation of a nucleus of arbitrary dimensions. | Solution. We regard the metastable phase as an external medium containing the nucleus, and calculate the work of formation of the nucleus from formula (20.2): \( {R}_{\min } = \Delta \left( {E - {T}_{0}S + {P}_{0}V}\right) \) or, since in this case the process occurs at constant temperature equal to the temperature of ... | Yes |
Distance, function to spline space (距离, 函数到样条空间) | Theorem on(定理), 368 | No |
Theorem 2. Suppose that\n\n(7)\n\n\[ \mathop{\lim }\limits_{{z \rightarrow {z}_{0}}}f\left( z\right) = {w}_{0}\;\text{ and }\;\mathop{\lim }\limits_{{z \rightarrow {z}_{0}}}F\left( z\right) = {W}_{0}. \]\n\nThen\n\n(8)\n\n\[ \mathop{\lim }\limits_{{z \rightarrow {z}_{0}}}\left\lbrack {f\left( z\right) + F\left( z\right... | This important theorem can be proved directly by using the definition of the limit of a function of a complex variable. But, with the aid of Theorem 1, it follows almost immediately from theorems on limits of real-valued functions of two real variables.\n\nTo verify property (9), for example, we write\n\n\[ f\left( z\r... | Yes |
Theorem 2. If a function \( f\left( z\right) \) is continuous and nonzero at a point \( {z}_{0} \), then \( f\left( z\right) \neq 0 \) throughout some neighborhood of that point. | Assuming that \( f\left( z\right) \) is, in fact, continuous and nonzero at \( {z}_{0} \), we can prove Theorem 2 by assigning the positive value \( \left| {f\left( {z}_{0}\right) }\right| /2 \) to the number \( \varepsilon \) in statement (4). This tells us that there is a positive number \( \delta \) such that\n\n\[ ... | Yes |
Theorem 3. If a function \( f \) is continuous throughout a region \( R \) that is both closed and bounded, there exists a nonnegative real number \( M \) such that\n\n\[ \left| {f\left( z\right) }\right| \leq M\;\text{ for all points }z\text{ in }R, \]\n\nwhere equality holds for at least one such \( z \) . | To prove this, we assume that the function \( f \) in equation (5) is continuous and note how it follows that the function\n\n\[ \sqrt{{\left\lbrack u\left( x, y\right) \right\rbrack }^{2} + {\left\lbrack v\left( x, y\right) \right\rbrack }^{2}} \]\n\n is continuous throughout \( R \) and thus reaches a maximum value \... | No |
Theorem 1. If a function \( f\left( z\right) = u\left( {x, y}\right) + {iv}\left( {x, y}\right) \) is analytic in a domain \( D \) , then its component functions \( u \) and \( v \) are harmonic in \( D \) . | To show this, we need a result that is to be proved in Chap. 4 (Sec. 52). Namely, if a function of a complex variable is analytic at a point, then its real and imaginary components have continuous partial derivatives of all orders at that point.\n\nAssuming that \( f \) is analytic in \( D \), we start with the observa... | Yes |
Theorem 2. A function \( f\left( z\right) = u\left( {x, y}\right) + {iv}\left( {x, y}\right) \) is analytic in a domain \( D \) if and only if \( v \) is a harmonic conjugate of \( u \) . | The proof is easy. If \( v \) is a harmonic conjugate of \( u \) in \( D \), the theorem in Sec. 22 tells us that \( f \) is analytic in \( D \) . Conversely, if \( f \) is analytic in \( D \), we know from Theorem 1 that \( u \) and \( v \) are harmonic in \( D \) ; furthermore, in view of the theorem in Sec. 21, the ... | No |
Theorem 1. If a function \( f \) is analytic at a given point, then its derivatives of all orders are analytic there too. | To prove this remarkable theorem, we assume that a function \( f \) is analytic at a point \( {z}_{0} \) . There must, then, be a neighborhood \( \left| {z - {z}_{0}}\right| < \varepsilon \) of \( {z}_{0} \) throughout which \( f \) is analytic (see Sec. 24). Consequently, there is a positively oriented circle \( {C}_{... | Yes |
Theorem 2. Let \( f \) be continuous on a domain D. If\n\n(1)\n\n\[{\int }_{C}f\left( z\right) {dz} = 0\]\n\nfor every closed contour \( C \) in \( D \), then \( f \) is analytic throughout \( D \). | To prove the theorem here, we observe that when its hypothesis is satisfied, the theorem in Sec. 44 ensures that \( f \) has an antiderivative in \( D \) ; that is, there exists an analytic function \( F \) such that \( {F}^{\prime }\left( z\right) = f\left( z\right) \) at each point in \( D \) . Since \( f \) is the d... | Yes |
Theorem 3. Suppose that a function \( f \) is analytic inside and on a positively oriented circle \( {C}_{R} \), centered at \( {z}_{0} \) and with radius \( R \) (Fig. 69). If \( {M}_{R} \) denotes the maximum value of \( \left| {f\left( z\right) }\right| \) on \( {C}_{R} \), then\n\n\[ \left| {{f}^{\left( n\right) }\... | Inequality (2) is called Cauchy's inequality and is an immediate consequence of the expression\n\n\[ {f}^{\left( n\right) }\left( {z}_{0}\right) = \frac{n!}{2\pi i}{\int }_{{C}_{R}}\frac{f\left( z\right) {dz}}{{\left( z - {z}_{0}\right) }^{n + 1}}\;\left( {n = 1,2,\ldots }\right) ,\]\n\nwhich is a slightly different fo... | Yes |
Theorem 1. If a function \( f \) is entire and bounded in the complex plane, then \( f\left( z\right) \) is constant throughout the plane. | To start the proof, we assume that \( f \) is as stated and note that since \( f \) is entire, Theorem 3 in Sec. 52 can be applied with any choice of \( {z}_{0} \) and \( R \) . In particular, Cauchy’s inequality (2) in that theorem tells us that when \( n = 1 \) ,\n\n(1)\n\n\[ \left| {{f}^{\prime }\left( {z}_{0}\right... | Yes |
Theorem 2. Any polynomial\n\n\\[ \nP\\left( z\\right) = {a}_{0} + {a}_{1}z + {a}_{2}{z}^{2} + \\cdots + {a}_{n}{z}^{n}\\;\\left( {{a}_{n} \\neq 0}\\right) \n\\]\n\nof degree \\( n\\left( {n \\geq 1}\\right) \\) has at least one zero. That is, there exists at least one point \\( {z}_{0} \\) such that \\( P\\left( {z}_{0... | The proof here is by contradiction. Suppose that \\( P\\left( z\\right) \\) is not zero for any value of \\( z \\) . Then the reciprocal\n\n\\[ \nf\\left( z\\right) = \\frac{1}{P\\left( z\\right) } \n\\]\n\nis clearly entire, and it is also bounded in the complex plane.\n\nTo show that its is bounded, we first write\n\... | Yes |
Corollary 1. If a series of complex numbers converges, the nth term converges to zero as \( n \) tends to infinity. | Assuming that series (1) converges, we know from the theorem that if\n\n\[ \n{z}_{n} = {x}_{n} + i{y}_{n}\;\left( {n = 1,2,\ldots }\right) , \n\]\n\nthen each of the series\n\n(8)\n\n\[ \n\mathop{\sum }\limits_{{n = 1}}^{\infty }{x}_{n}\;\text{ and }\;\mathop{\sum }\limits_{{n = 1}}^{\infty }{y}_{n} \n\]\nconverges. We... | Yes |
Corollary 2. The absolute convergence of a series of complex numbers implies the convergence of that series. | To prove Corollary 2, we assume that series (1) converges absolutely. Since\n\n\[ \left| {x}_{n}\right| \leq \sqrt{{x}_{n}^{2} + {y}_{n}^{2}}\;\text{ and }\;\left| {y}_{n}\right| \leq \sqrt{{x}_{n}^{2} + {y}_{n}^{2}}, \]\n\nwe know from the comparison test in calculus that the two series\n\n\[ \mathop{\sum }\limits_{{n... | Yes |
Theorem 1. If a power series\n\n\[ \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{\left( z - {z}_{0}\right) }^{n} \]\n\nconverges when \( z = {z}_{1}\left( {{z}_{1} \neq {z}_{0}}\right) \), then it is absolutely convergent at each point \( z \) in the open disk \( \left| {z - {z}_{0}}\right| < {R}_{1} \) where \( {R}... | We start the proof by assuming that the series\n\n\[ \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{\left( {z}_{1} - {z}_{0}\right) }^{n}\;\left( {{z}_{1} \neq {z}_{0}}\right) \]\n\nconverges. The terms \( {a}_{n}{\left( {z}_{1} - {z}_{0}\right) }^{n} \) are thus bounded; that is,\n\n\[ \left| {{a}_{n}{\left( {z}_{1}... | Yes |
Theorem 2. If \( {z}_{1} \) is a point inside the circle of convergence \( \left| {z - {z}_{0}}\right| = R \) of a power series\n\n(4)\n\n\[ \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{\left( z - {z}_{0}\right) }^{n} \]\n\nthen that series must be uniformly convergent in the closed disk \( \left| {z - {z}_{0}}\rig... | Our proof of this theorem depends on Theorem 1. Given that \( {z}_{1} \) is a point lying inside the circle of convergence of series (4), we note that there are points inside that circle and farther from \( {z}_{0} \) than \( {z}_{1} \) for which the series converges. So, according to Theorem 1,\n\n(5)\n\n\[ \mathop{\s... | Yes |
Theorem 1. Let \( C \) denote any contour interior to the circle of convergence of the power series (1), and let \( g\left( z\right) \) be any function that is continuous on \( C \) . The series formed by multiplying each term of the power series by \( g\left( z\right) \) can be integrated term by term over \( C \) ; t... | To prove this theorem, we note that since both \( g\left( z\right) \) and the sum \( S\left( z\right) \) of the power series are continuous on \( C \), the integral over \( C \) of the product\n\n\[ \ng\left( z\right) S\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{{N - 1}}{a}_{n}g\left( z\right) {\left( z - {z}_{0... | Yes |
Theorem 2. The power series (1) can be differentiated term by term. That is, at each point \( z \) interior to the circle of convergence of that series,\n\n(6)\n\n\[ \n{S}^{\prime }\left( z\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }n{a}_{n}{\left( z - {z}_{0}\right) }^{n - 1}.\n\] | To prove this, let \( z \) denote any point interior to the circle of convergence of series (1). Then let \( C \) be some positively oriented simple closed contour surrounding \( z \) and interior to that circle. Also, define the function\n\n(7)\n\n\[ \ng\left( s\right) = \frac{1}{2\pi i} \cdot \frac{1}{{\left( s - z\r... | Yes |
Theorem 1. If a series\n\n\[ \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{\left( z - {z}_{0}\right) }^{n} \]\n\nconverges to \( f\left( z\right) \) at all points interior to some circle \( \left| {z - {z}_{0}}\right| = R \), then it is the Taylor series expansion for \( f \) in powers of \( z - {z}_{0} \) . | To start the proof, we write the series representation\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{\left( z - {z}_{0}\right) }^{n}\;\left( {\left| {z - {z}_{0}}\right| < R}\right) \]\n\nin the hypothesis of the theorem using the index of summation \( m \) :\n\n\[ f\left( z\right) = \mathop... | Yes |
Theorem 2. If a series\n\n\[ \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{c}_{n}{\left( z - {z}_{0}\right) }^{n} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{\left( z - {z}_{0}\right) }^{n} + \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{b}_{n}}{{\left( z - {z}_{0}\right) }^{n}} \]\n\nconverges to \( f\lef... | The method of proof here is similar to the one used in proving Theorem 1. The hypothesis of this theorem tells us that there is an annular domain about \( {z}_{0} \) such that\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{c}_{n}{\left( z - {z}_{0}\right) }^{n} \]\n\nfor each point \( z \) i... | Yes |
Theorem 2. Given a function \( f \) and a point \( {z}_{0} \), suppose that\n\n(a) \( f \) is analytic at \( {z}_{0} \) ;\n\n(b) \( f\left( {z}_{0}\right) = 0 \) but \( f\left( z\right) \) is not identically equal to zero in any neighborhood of \( {z}_{0} \) .\n\nThen \( f\left( z\right) \neq 0 \) throughout some delet... | To prove this, let \( f \) be as stated and observe that not all of the derivatives of \( f \) at \( {z}_{0} \) are zero. If they were, all of the coefficients in the Taylor series for \( f \) about \( {z}_{0} \) would be zero; and that would mean that \( f\left( z\right) \) is identically equal to zero in some neighbo... | Yes |
Theorem 3. Given a function \( f \) and a point \( {z}_{0} \), suppose that\n\n(a) \( f \) is analytic throughout a neighborhood \( {N}_{0} \) of \( {z}_{0} \) ;\n\n(b) \( f\left( z\right) = 0 \) at each point \( z \) of a domain \( D \) or line segment \( L \) containing \( {z}_{0} \) (Fig. 90).\n\nThen \( f\left( z\r... | We begin the proof with the observation that under the stated conditions, \( f\left( z\right) \equiv 0 \) in some neighborhood \( N \) of \( {z}_{0} \) . For, otherwise, there would be a deleted neighborhood of \( {z}_{0} \) throughout which \( f\left( z\right) \neq 0 \), according to Theorem 2 ; and that would be inco... | Yes |
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