A course of modern analysis

Cover......Page 0001.djvu
A Course of Modern Analysis......Page 0006.djvu
ISBN 0521588073......Page 0007.djvu
Preface......Page 0008.djvu
Contents......Page 0010.djvu
PART I. THE PROCESSES OF ANALYSIS......Page 0012.djvu
1.1. Rational numbers......Page 0014.djvu
1.2. Dedekind\'s theory of irrational numbers......Page 0015.djvu
1.3. Complex numbers......Page 0017.djvu
1.4. The modulus of a complex number......Page 0019.djvu
Miscellaneous Examples......Page 0021.djvu
2.11. Definition of the phrase \'of the order of\'......Page 0022.djvu
2.21. Limit-points and the Bolzano-Weierstrass theorem......Page 0023.djvu
2.22. Cauchy\'s theorem on the necessary and sufficient condition for the existance of a limit......Page 0024.djvu
2.30......Page 0026.djvu
2.301. Abel\'s inequality......Page 0027.djvu
2.31. Dirichlet\'s test for convergence......Page 0028.djvu
2.33. The geometric series, and the series sum(1^s)......Page 0029.djvu
2.34. The Comparison Theorem......Page 0031.djvu
2.35. Cauchy\'s test for absolute convergence......Page 0032.djvu
2.36. D\'Alembert\'s ratio test for absolute convergence......Page 0033.djvu
2.37. A general theorem on series  which lim|u_(n+1)/u_n| = 1......Page 0034.djvu
2.38. Convergence of the hypergeometric series......Page 0035.djvu
2.41 The fundamental property of absolutely convergent series......Page 0036.djvu
2.5. Double series......Page 0037.djvu
2.51. Methods of summing double series......Page 0038.djvu
2.52. Absolutely convergent double series......Page 0039.djvu
2.6. Power-Series......Page 0040.djvu
2.61. Convergence of series derived from a power-series......Page 0042.djvu
2.7. Infinite Products......Page 0043.djvu
2.71. Some examples of infinite products......Page 0044.djvu
2.81. Convergence of an infinite determinant......Page 0047.djvu
2.82. The rearrangement Theorem for convergent infinite determinants......Page 0048.djvu
Miscellaneous Examples......Page 0049.djvu
3.2. Continuity of functions of real variables......Page 0052.djvu
3.21. Simple curves. Continua......Page 0054.djvu
3.3. Series of variable terms. Uniformity of convergence......Page 0055.djvu
3.31. On the condition for uniformity of convergence......Page 0056.djvu
3.32. Connexion of discontinuity with non-uniform convergence......Page 0058.djvu
3.33. The distinction between absolute and uniform convergence......Page 0059.djvu
3.341. Uniforrnity of convergence of infinite products......Page 0060.djvu
3.35. Hardy\'s tests for uniform convergence......Page 0061.djvu
3.4. Discussion of a particular double series......Page 0062.djvu
3.5. The concept of uniformity......Page 0063.djvu
3.6. The modified Heine-Borel theorem......Page 0064.djvu
3.61. Uniformity of continuity......Page 0065.djvu
3.62. A real function, of a real variable, continuous in a closed interval, attains its upper bound......Page 0066.djvu
3.64. The fluctuation of a function of a real variable......Page 0067.djvu
3.71. Abel\'s theorem on continuity up to the circle of convergence......Page 0068.djvu
3.73. Power series which vanish identically......Page 0069.djvu
Miscellaneous Examples......Page 0070.djvu
4.11. Upper and lower integrals......Page 0072.djvu
4.13. A general theorem on integration......Page 0074.djvu
4.14. Mean Value Theorems......Page 0076.djvu
4.2. Differentiation of integrals containing a parameter......Page 0078.djvu
4.3. Double integrals and repeated integrals......Page 0079.djvu
4.4. Infinite integrals......Page 0080.djvu
4.43. Tests for the convergence of an infinite integral......Page 0081.djvu
4.431. Tests for uniformity of convergence of an infinite integral......Page 0083.djvu
4.44. Theorems concerning uniformly convergent infinite integrals......Page 0084.djvu
4.51. The inversion of the order of integration of a certain repeated integral......Page 0086.djvu
4.6. Complex integration......Page 0088.djvu
4.7. Integration of infinite series......Page 0089.djvu
References......Page 0091.djvu
Miscellaneous Examples......Page 0092.djvu
V The fundamental properties of Analytic Functions; Taylor\'s, Laurent\'s,and Liouville\'s Theorems......Page 0093.djvu
5.4. Taylor\'s theorem......Page 0104.djvu
5.12. Cauchy\'s definition of an analytic function of a complex variable......Page 0094.djvu
5.13. An application of the modified Heine-Borel theorem......Page 0095.djvu
5.2. Cauchy\'s theorem on the integral of a function round a contour......Page 0096.djvu
5.21. The value of a function at a point, expressed as an integral taken round a contour enclosing the point......Page 0099.djvu
5.22. The derivates of an analytic function f(z)......Page 0100.djvu
5.3. Analytic functions represented by uniformly convergent series......Page 0102.djvu
5.32. Analytic functions represented by infinite integrals......Page 0103.djvu
5.41. Forms of the remainder in Taylor\'s series......Page 0106.djvu
5.5. The process of continuation......Page 0107.djvu
5.51. The identity of two functions......Page 0109.djvu
5.6. Laurent\'s theorem......Page 0111.djvu
5.61. The nature of the singularities of a one-valued function......Page 0113.djvu
5.62. The \'point at infinity\'......Page 0114.djvu
5.64. Functions with no essential singularities......Page 0116.djvu
5.7. Many-valued functions......Page 0117.djvu
Miscellaneous Examples......Page 0119.djvu
6.1. Residues......Page 0122.djvu
6.21. The evaluation of the integrals of certain periodic functions taken between the limits 0 and 2π......Page 0123.djvu
6.22. The evaluation of certain types of integrals taken between the limits -∞ and +∞......Page 0124.djvu
6.221. Certain infinite integrals involving sines and cosines......Page 0125.djvu
6.222. Jordan\'s lemma......Page 0126.djvu
6.24. Evaluation of integrals of the form ∫ [0, ∞] x^(a-1)Q(x)dx......Page 0128.djvu
6.31. The number of roots of an equation contained within a contur......Page 0130.djvu
References......Page 0132.djvu
Miscellaneous Examples......Page 0133.djvu
7.2. The Bernoullian numbers and the Bernoullian polynomials......Page 0136.djvu
7.21. The Euler-Maclaurin expansion......Page 0138.djvu
7.3. Bürmann\'s theorem......Page 0139.djvu
7.31. Teixeira\'s extended form of Bürmann\'s theorem......Page 0142.djvu
7.32. Lagrange\'s theorem......Page 0143.djvu
7.4. The expansion of a class of functions in rational fractions......Page 0145.djvu
7.5. The expansion of a class of functions as infinite products......Page 0147.djvu
7.6. The factor theorem of Weierstrass......Page 0148.djvu
7.7. The expansion of a class of periodic functions in a series of cotangents......Page 0150.djvu
7.8. Borel\'s therem......Page 0151.djvu
7.81. Borel\'s integral and analytic continuation......Page 0152.djvu
7.82. Expansions in series of inverse factorials......Page 0153.djvu
Miscellaneous Examples......Page 0155.djvu
8.1. Simple example of an asymptotic expansion......Page 0161.djvu
8.21. Another example of an asymptotic expansion......Page 0162.djvu
8.3. Multiplication of asymptotic expansions......Page 0163.djvu
8.32. Uniqueness of an asymptotic expansion......Page 0164.djvu
8.41. Borel\'s method of summation......Page 0165.djvu
8.43. Cesàro\'s method of summation......Page 0166.djvu
8.5. Hardy\'s convergence theorem......Page 0167.djvu
Miscellaneous Examples......Page 0170.djvu
9.1. Definition of Fourier series......Page 0171.djvu
9.11. Nature of the region within which a trigonometrical series converges......Page 0172.djvu
9.2. On Dirichlet\'s conditions and Fourier\'s theorem......Page 0174.djvu
9.22. The cosine series and the sine series......Page 0176.djvu
9.3. The nature of the coefficients in a Fourier series......Page 0178.djvu
9.31. Differentiation of Fourier series......Page 0179.djvu
9.4. Fejér\'s theorem......Page 0180.djvu
9.41. The Riemann-Lebesgue lemmas......Page 0183.djvu
9.42. The proof of Fourier\'s theorem......Page 0186.djvu
9.43. The Dirichlet-Bonnet proof of Fourier\'s theoem......Page 0187.djvu
9.44. The uniformity of the convergence of Fourier series......Page 0190.djvu
9.5. The Hurwitz-Liapounoff theorem concerning Fourier constants......Page 0191.djvu
9.6. Riemann\'s theory of trigonometrical series......Page 0193.djvu
9.61. Riemann\'s associated function......Page 0194.djvu
9.62. Properties of Riemann\'s associated function; Riemann\'s first lemma......Page 0195.djvu
9.621. Riemann\'s second lemma......Page 0196.djvu
9.631. Schwartz\' lemma......Page 0197.djvu
9.632. Proof of Riemann\'s Theorem......Page 0198.djvu
9.7. Fourier\'s representation of a function by an integral......Page 0199.djvu
References......Page 0200.djvu
Miscellaneous Examples......Page 0201.djvu
10.2. Solution of a differential equation valid in the vicinity of an ordinary point......Page 0205.djvu
10.21. Uniqueness of the solution......Page 0207.djvu
10.3. Points which are regular for a differential equation......Page 0208.djvu
10.31. Convergence of the expansion of Section 10.3.......Page 0210.djvu
10.32. Derivation of a second solution in the case when the difference of the exponents is an integer or zero......Page 0211.djvu
10.5. Irregular singularities and confluence......Page 0213.djvu
10.6. The differential equations of mathematical physics......Page 0214.djvu
10.7. Linear differential equations with three singularities......Page 0217.djvu
10.71. Transforrnations of Riemann\'s P-equation......Page 0218.djvu
References......Page 0219.djvu
Miscellaneous Examples......Page 0220.djvu
11.1. Definition of an integral equation......Page 0222.djvu
11.11. An algebraical lemma......Page 0223.djvu
11.2. Fredholm\'s equation and its tentative solution......Page 0224.djvu
11.21. Investigation of Fredholm\'s solution......Page 0226.djvu
11.22. Volterra\'s reciprocal functions......Page 0229.djvu
11.23. Homogeneous integral equations......Page 0230.djvu
11.4. The Liouville-Neumann method of successive substitutions......Page 0232.djvu
11.51. Schmidt\'s theorem that, if the nucleus is symmetric, the equation D(λ) = 0 has at least one root......Page 0234.djvu
11.6. Orthogonal functions......Page 0235.djvu
11.61. The connexion of orthogonal functions with homogeneous integral equations......Page 0236.djvu
11.7. The development of a symmetric nucleus......Page 0238.djvu
11.71. The solution of Fredholm\'s equation by a series......Page 0239.djvu
11.81. Schlömilch\'s integral equation......Page 0240.djvu
Miscellaneous Examples......Page 0241.djvu
PART II. THE TRANSCENDENTAL FUNCTIONS......Page 0244.djvu
12.1. Definitions of the Gamma-function, The Weierstrassian product......Page 0246.djvu
12.12. The difference equation satisfied by the Gamma-function......Page 0248.djvu
12.13. The evaluation of a general class of infinite products......Page 0249.djvu
12.14. Connexion between the Gamma-function and the circular functions......Page 0250.djvu
12.15. The multiplication-theorem of Gauss and Legendre......Page 0251.djvu
12.2. Euler\'s expression of Γ(z) as an infinite integral......Page 0252.djvu
12.21. Extension of the Infinite integral to the case in which the argumnent of the Gamma-function is negative......Page 0254.djvu
12.22. Hankel\'s expression of Γ(z) as a contour integral......Page 0255.djvu
12.3. Gauss\' expression for the logarithmic derivate of the Gamma-function as an infinite integral......Page 0257.djvu
12.31. Binet\'s first expression for log Γ(z) in terms of an infinite integral......Page 0259.djvu
12.32. Binet\'s second expression for log Γ(z) in terms of an infinite integral......Page 0261.djvu
12.33. The Asymptotic Expansion of the Logarithm of the Gamma-Function (Stirling\'s series)......Page 0262.djvu
12.4. The Eulerian Integral of the First Kind......Page 0264.djvu
12.41. Expression of the Eulerian Integral of the First Kind in terms of the Gamma-function......Page 0265.djvu
12.43. Pochhammer\'s extension of the Eulerian Integral of the First Kind......Page 0267.djvu
12.5. Dirichlet\'s integral......Page 0269.djvu
Miscellaneous Examples......Page 0270.djvu
13.12. The expression of  ζ(s, a) as an infinite integral......Page 0276.djvu
13.13. The expression of  ζ(s, a) as a contour integral......Page 0277.djvu
13.14. Values of ζ(s, a) for special values of s......Page 0278.djvu
13.15. The formula of Hurwitz for ζ(s, a) when σ < 0......Page 0279.djvu
13.2. Hermite\'s formula for  ζ(s, a)......Page 0280.djvu
13.21. Deductions from Hermite\'s formula......Page 0282.djvu
13.31. Riemann\'s hypothesis concerning the zeros of  ζ(s)......Page 0283.djvu
13.4. Riemann\'s integral for ζ(s)......Page 0284.djvu
13.5. Inequalities satisfied by ζ(s, a) when σ > 0......Page 0285.djvu
13.51. lnequalities satisfied by ζ(s, a) when σ ≤ 0......Page 0286.djvu
13.6. The asymptotic expansion of log Γ(z + a)......Page 0287.djvu
Miscellaneous Examples......Page 0290.djvu
14.11. The value of F(a, b; c; 1) when R(c - a - b) > 0......Page 0292.djvu
14.3. Solutions of Riemann\'s P-equation by hypergeometric functions......Page 0294.djvu
14.4. Relations between particular solutions of the hypergeometric equation......Page 0296.djvu
14.5. Barnes\' contour integrals for the hypergeometric function......Page 0297.djvu
14.51. The continuation of the hypergeometric series......Page 0299.djvu
14.52. Barnes\' lemma......Page 0300.djvu
14.53. The connexion between hypergeometric functions of z and of 1 - z......Page 0301.djvu
14.6. Solution of Riemann\'s equation by a contour integral......Page 0302.djvu
14.61. Determination of an integral which represents P^(a)......Page 0304.djvu
14.7. Relations between contiguous hypergeometric functions......Page 0305.djvu
Miscellaneous Examples......Page 0307.djvu
15.1. Definition of Legendre polynomials......Page 0313.djvu
15.12. Schläfli\'s integral for P_n(z)......Page 0314.djvu
15.13. Legendre\'s differential equation......Page 0315.djvu
15.14. The integral properties of the Legendre polynomials......Page 0316.djvu
15.2. Legendre functions......Page 0317.djvu
15.21. The Recurrence Formulae......Page 0318.djvu
15.211. The expression of any polynomial as a series of Legendre polynomials......Page 0321.djvu
15.22. Murphy\'s expression of P_n(z) as a hypergeometric function......Page 0322.djvu
15.231. The Mehler-Dirichlet integral for P_n(z)......Page 0323.djvu
15.3. Legendre functions of the second kind......Page 0327.djvu
15.31. Expansion of Q_n(z) as a power-series......Page 0328.djvu
15.32. The recurrence-formulae for Q_n(z)......Page 0329.djvu
15.33. The Laplacian integral for Legendre functions of the second kind......Page 0330.djvu
15.34. Neumann\'s formula far Q_n(z), when n is an integer......Page 0331.djvu
15.4. Heine\'s development of (t - Z)^(-1) as a series of Legendre polynomials in z......Page 0332.djvu
15.41. Neumann\'s expansion of an arbitrary function in a series of Legendre polynomials......Page 0333.djvu
15.5. Ferrers associated Legendre functions P^m_n(z) and Q^m_n(z)......Page 0334.djvu
15.51. The integral properties of the associated Legendre functions......Page 0335.djvu
15.6. Hobson\'s definition of the associated Legendre functions......Page 0336.djvu
15.7. The addition theorem for the Legendre polynomials......Page 0337.djvu
15.71. The addition theorem for the Legendre functions......Page 0339.djvu
15.8. The function C^ν_n(z)......Page 0340.djvu
Miscellaneous Examples......Page 0341.djvu
16.1. The confluence of two singularities of Riemann\'s equation......Page 0348.djvu
16.11. Kummer\'s formulae......Page 0349.djvu
16.12. Definition of the function W_k,_m(z)......Page 0350.djvu
16.2. Expression of various functions by functions of the type W_k,_m(z)......Page 0351.djvu
16.3. The asymptotic expansion of W_k,_m(z), when |z| is large......Page 0353.djvu
16.4. Contour integrals of the Mellin-Barnes type for W_k,_m(z)......Page 0354.djvu
16.51. The second solution of Weber\'s equation......Page 0358.djvu
16.52. The general asymptotic expansion of D_n(z)......Page 0359.djvu
16.6. A contour integral for D_n(z)......Page 0360.djvu
16.7. Properties of D_n(z) when n is an integer......Page 0361.djvu
Miscellaneous Examples......Page 0362.djvu
17.1. The Bessel coefficients......Page 0366.djvu
17.11. Bessel\'s differential equation......Page 0368.djvu
17.2. The solution of Bessel\'s equation when n is not necessarily an integer......Page 0369.djvu
17.21. The recurrence formulae for the Bessel functions......Page 0370.djvu
17.212. The connexion between, J+_n(z) and W_k,_m functions......Page 0371.djvu
17.22. The zeros of Bessel functions whose order n is real......Page 0372.djvu
17.231. The modification of Bessel\'s integral when n is not an integer......Page 0373.djvu
17.24. Bessel functions whose order is half an odd integer......Page 0375.djvu
17.3. Hankel\'s contour integral for J_n(z)......Page 0376.djvu
17.4. Connexion between Bessel coefficients and Legendre functions......Page 0378.djvu
17.5. Asymptotic series for J_n(z) when |Z| is large......Page 0379.djvu
17.6. The second solution of Bessel\'s equation when the order is an integer......Page 0380.djvu
17.61. The ascending series for Y_n(z)......Page 0382.djvu
17.7. Bessel functions with purely imaginary argument......Page 0383.djvu
17.71. Modified Bessel functions of the second kind......Page 0384.djvu
17.8. Neumann\'s expansion of an analytic function in a series of Bessel coefficients......Page 0385.djvu
17.81. Proof of Neumann\'s expansion......Page 0386.djvu
17.82. Schlomilch\'s expansion of an arbitrary function in a series of Bessel coefficients of order zero......Page 0388.djvu
References......Page 0389.djvu
Miscellaneous Examples......Page 0390.djvu
18.1. The differential equations of mathematical physics......Page 0397.djvu
18.2. Boundary conditions......Page 0398.djvu
18.3. A general solution of Laplace\'s equation......Page 0399.djvu
18.31. Solutions of Laplace\'s equation involving Legendre functions......Page 0402.djvu
18.4. The solution of Laplace\'s equation which satisfies assigned boundary conditions at the surface of a sphere......Page 0404.djvu
18.5. Solutions of Laplace\'s equation which involve Bessel coefficients......Page 0406.djvu
18.51. The periods of vibration of a uniform membrane......Page 0407.djvu
18.61. Solutions of the equation of wave motions which involve Bessel functions......Page 0408.djvu
Miscellaneous Examples......Page 0410.djvu
19.1. The differential equation of Mathieu......Page 0415.djvu
19.12. Hill\'s equation......Page 0417.djvu
19.21. An integral equation satisfied by even Mathieu functiuns......Page 0418.djvu
19.22. Proof that the even Mathieu functions satisfy the integral equatian......Page 0419.djvu
19.3. The construction of Mathieu functions......Page 0420.djvu
19.31. The integral formulae for the Mathieu functions......Page 0422.djvu
19.4. The nature of the solution of Mathieu\'s general equation; Floquet\'s theory......Page 0423.djvu
19.41. Hill\'s method of solution......Page 0424.djvu
19.42. The evaluation of Hill\'s determinant......Page 0426.djvu
19.51. Lindemann\'s form of Floqnet\'s theorem......Page 0428.djvu
19.52. The determination of the integral function associated with Mathieu\'s general equation......Page 0429.djvu
19.53. The solution of Mathieu\'s equation in terms of F(ζ)......Page 0430.djvu
19.6. A second method of constructing the Mathieu function......Page 0431.djvu
19.7. The method of change of parameter......Page 0435.djvu
19.8. The asymptotic solution of Mathieu\'s equation......Page 0436.djvu
Miscellaneous Examples......Page 0437.djvu
20.1. Doubly-periodic functions......Page 0440.djvu
20.11. Period-parallelograms......Page 0441.djvu
20.12. Simple properties of elliptic functions......Page 0442.djvu
20.13. The order of an elliptic function......Page 0443.djvu
20.2. The construction of an elliptic function. Definition of P(z)......Page 0444.djvu
20.21. Periodicity and other properties of P(z)......Page 0445.djvu
20.22. The differential equation satisfied by P(z)......Page 0447.djvu
20.221. The integral formula for P(z)......Page 0448.djvu
20.222. An illustration from the theory of the circular functions......Page 0449.djvu
20.31. Another form of the addition-theorem......Page 0451.djvu
20.311. The duplication formula for P(z)......Page 0452.djvu
20.32. The constants e₁, e₂, e₃......Page 0454.djvu
20.33. The addition of a half-period to the argument of P(z)......Page 0455.djvu
20.4. Quasi-periodic functions. The function ζ(z)......Page 0456.djvu
20.411. The relation between ν₁ and ν₂......Page 0457.djvu
20.421. The quasi-periodicity of the function σ(z)......Page 0458.djvu
20.51. The expression of any elliptic function in terms of P(z) and  P\'(z)......Page 0459.djvu
20.52. The expression of any elliptic function as a linear combination of Zeta-functions and their derivates......Page 0460.djvu
20.53. The expression of any elliptic function as a quotient of Sigma-functions......Page 0461.djvu
20.6. On the integration 0f {a₀x⁴ + 4a₁x³ + 6a₂x² + 4a₃x + a₄ }^-(1/2)......Page 0463.djvu
20.7. The uniformisation of curves of genus unity......Page 0465.djvu
References......Page 0466.djvu
Miscellaneous Examples......Page 0467.djvu
21.1. The definition of a Theta-function......Page 0473.djvu
21.11. The four types of Theta-functions......Page 0474.djvu
21.12. The zeros of the Theta-functions......Page 0476.djvu
21.2. The relations between the squares of the Theta-functions......Page 0477.djvu
21.22. Jacobi\'s fundamental formulae......Page 0478.djvu
21.3. Jacobi\'s expressions for the Theta-functiuns as infinite products......Page 0480.djvu
21.41. A relation between Theta-functions of zero argument......Page 0481.djvu
21.42. The value of the constant G......Page 0483.djvu
21.5. The expression of elliptic functions by means of Theta-functions......Page 0484.djvu
21.51. Jacobi\'s imaginary transformation......Page 0485.djvu
21.52. Landen\'s type of transformation......Page 0487.djvu
21.6. The differential equations satisfied by quotients of Theta-functions......Page 0488.djvu
21.61. The genesis of the Jacobian Elliptic function sn u......Page 0489.djvu
21.62. Jacobi\'s earlier notation. The Theta-function Θ(u) and the Eta-function H(u)......Page 0490.djvu
21.7. The problem of Inversion......Page 0491.djvu
21.71. The problem of inversion for complex values of c. The modular functions f(τ), g(τ), h(τ)......Page 0492.djvu
21.711. The principal solution of f(τ) - C = 0......Page 0493.djvu
21.73. The inversion-problem associated with Weierstrassian elliptic functions......Page 0495.djvu
21.8. The numerical computation of elliptic functions......Page 0496.djvu
Miscellaneous Examples......Page 0498.djvu
22.11. The Jacobian elliptic functions, sn u, cn u, dn u......Page 0502.djvu
22.121. The complementary modulus......Page 0504.djvu
22.2. The addition-theorem for the function sn u......Page 0505.djvu
22.21. The addition-theorems for cn u and dn u......Page 0508.djvu
22.30......Page 0509.djvu
22.302. The equivilence of the definitions of K......Page 0510.djvu
22.31. The periodic properties (associated with K) of the Jacobian elliptic functions......Page 0511.djvu
22.32. The constant K\'......Page 0512.djvu
22.33. The periodic properties (associated with K + iK\') of the Jacobian elliptic functions......Page 0513.djvu
22.341. The behaviour of the Jacobian elliptic functions near the origin and near iK\'......Page 0514.djvu
22.35. General description of the functions sn u, cn u, dn u......Page 0515.djvu
22.4. Jacobi\'s imaginary transformation......Page 0516.djvu
22.41 Proof of Jacobi\'s iimaginary transformation by the aid of Theta-functions......Page 0517.djvu
22.42. Landen\'s transfornation......Page 0518.djvu
22.5. Infinite products for the Jacobian elliptic functions......Page 0519.djvu
22.6. Fourier series for the Jacobian elliptic functions......Page 0521.djvu
22.61. Fourier series for reciprocals of Jacobian elliptic functions......Page 0522.djvu
22.7. Elliptic integrals......Page 0523.djvu
22.71. The expression of a quartic as the product of sums of squares......Page 0524.djvu
22.72. The three kinds of elliptic integrals......Page 0525.djvu
22.73. The elliptic integral of the second kind. The function E(u)......Page 0528.djvu
22.732. The addition-formulae for E(u) and Z(u)......Page 0529.djvu
22.734. Jacobi\'s imaginary transformation of E(u)......Page 0530.djvu
22.735. Legendre\'s relation......Page 0531.djvu
22.737. The values of the complete elliptic integrals for small values of k......Page 0532.djvu
22.74. The elliptic integral of the third kind......Page 0533.djvu
22.741. A dynamical application of the elliptic integral of the third kind......Page 0534.djvu
22.8. The lemniscate functions......Page 0535.djvu
22.81. The values of K and K\' for special values of k......Page 0536.djvu
22.82. A geometrical illustration of the functions sn u, en u, dn u......Page 0538.djvu
Miscellaneous Examples......Page 0539.djvu
23.1. The definition of ellipsoidal harmnonics......Page 0547.djvu
23.2. The four species of ellipsoidal harmonics......Page 0548.djvu
23.21. The construction of ellipsoidal harmonics of the first species......Page 0549.djvu
23.22. Ellipsoidal harmonics of the second species......Page 0551.djvu
23.23. Ellipsoidal harmonics of the third species......Page 0552.djvu
23.24. Ellipsoidal harmonics of the fourth species......Page 0553.djvu
23.25. Niven\'s expressions for ellipsoidal harmonics in terms of homogeneous harmonics......Page 0554.djvu
23.26. Ellipsoidal harmonics of degree n......Page 0557.djvu
23.3. Confocal coordinates......Page 0558.djvu
23.31. Uniformising variables associated with confocal coordinates......Page 0560.djvu
23.32. Laplace\'s equation referred to confocal coordinate......Page 0562.djvu
23.33 Ellipsoidal harmonics referred to confocal coordinates......Page 0563.djvu
23.4. Various forms of Lamé\'s differential equation......Page 0565.djvu
23.41. Solutions in series of Lamé\'s equation......Page 0567.djvu
23.43. The non-repetition of factors in Lamé functions......Page 0569.djvu
23.44. The linear independence of Lamé functions......Page 0570.djvu
23.46. Stieltjes\' theorem on the zeros of Lamé functions......Page 0571.djvu
23.47. Lamé functions of the second kind......Page 0573.djvu
23.5. Lamé\'s equation in association with Jacobian elliptic functions......Page 0574.djvu
23.6. The integral equation satisfied by Lamé functions of the first and second species......Page 0575.djvu
23.61. The integral equation satisfied by Lamé functions of the third and fourth species......Page 0577.djvu
23.62. Integral formulae for ellipsoidal harmonics......Page 0578.djvu
23.63. Integral formulae for ellipsoidal harmonics of the third and fourth speces......Page 0580.djvu
23.7. Generalisations of Lamé\'s equation......Page 0581.djvu
23.71. The Jacobian form of the generalised Lamé equation......Page 0584.djvu
References......Page 0586.djvu
Miscellaneous Examples......Page 0587.djvu
APPENDIX......Page 0590.djvu
LIST OF AUTHORS QUOTED......Page 0602.djvu
GENERAL INDEX......Page 0606.djvu