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| Macrobert T.M. — Functions of a complex variable |
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| Предметный указатель |
Legendre’s associated equation 240 259
Legendre’s associated functions 250 259—265 334 335 339
Legendre’s associated functions, as definite integrals 260—263 308 333 336 337
Legendre’s associated functions, asymptotic expansions of 303—307 309
Legendre’s associated functions, recurrence formulae for 310—314 337 338
Legendre’s associated functions, relations between 251 262—265 275 276 303 306 309 317
Legendre’s complete elliptic integrals of the first and second kinds 174
Legendre’s equation 213 234
Legendre’s equation, relation to Gauss’s equation 235
Legendre’s first normal elliptic integral 163 173
Legendre’s first normal elliptic integral, inversion of 166 201
Legendre’s normal integrals 173
Legendre’s relation 175 188
LIMIT 22
Limit, at infinity 22
Limit, infinite 23
Limit, of a sequence 42
Limit, of function, geometrical illustration 22
Limit, of quotient of two functions 30 83 295
Limit, uniform convergency to a 23
Liouville’s theorem 68
Logarithmic function 34—36 83 161
Logarithmic transformation 35 325
Loops 145 162 164 168
Loops, about point at infinity 162 168
Loops, notation for negative 162
Maclaurin’s expansion 296
Mehler — Dirichlet integrals 112 336 337
Mittag — Leffler’s theorem 105
Modulus, of complex number 2 3 4;
Multiplication of complex numbers 1 4
Naperian logarithms 34
Numbers, complex, imaginary, real 1 2
Numbers, complex, imaginary, real, geometrical representation of 1—6
Orthogonal systems 32
P-function, Riemann’s 244
Path of variation 7 10 22
Period, of a function 86 179
Period, of a function, parallelogram 179
Period, of a function, primitive 86 179
Point at infinity 9 38 39
Points of inflection on cubic 197
Points, congruent 179 180
Points, critical 38
Points, ordinary 38 210
Points, singular 38
Pole 38 39 67 118
Pole, an isolated singularity 39
Pole, at infinity 38 88
Pole, of order  38 86
Pole, principal part at a 86
Pole, simple 38
Polynomials 88
Power, the generalised 36
Product infinite 107 108
Product infinite, expression of function as 108 109
Product infinite, zeros of 295
Quantities  ,  , positive 23
R( ) notation 1
Region, closed 92
Region, connected 30
Region, function holomorphic in 53
Region, multiply-connected 30 47 58
Region, of existence of function 7
Region, of uniform convergence 92 96
Region, simply-connected 30
Remainder in Maclaurin’s expansion 296
Residue at a pole 57 58 67 96 295
Residue at a pole, at infinity 58 96
Riemann’s P-function 244
Riemann’s P-function, in terms of hypergeometric functions 246
Riemann’s P-function, indices of 245
Rodrigues’ formula 120
Root extraction 1 5 36
Roots of equations 4 5 16 69 378 400
Roots of equations, theorems on 118 119
Rouch ’s theorem 400
Routh’s condition 378
| Saalsch tz’s theorem 360
Saalschtz’s theorem, generalisation of 365
SEQUENCE 42 359
Series, convergent 76
Series, multiplication of 77 82
Series, power 80 82 95 125
Series, uniformly convergent 92
Series, well-poised 366
Sigma functions 109
Sigma functions, duplication formula for 190
Sigma functions, elliptic functions in terms of 190
Sigma functions, properties of 189
Similar figures 8 37
Singularities 38
Singularities, at infinity 38 39 88 89 106 181
Singularities, essential 39 86 89 90 106 181
Singularities, isolated 38 39
Singularities, line of 101
Singularities, non-essential 39
Singularities, of a differential equations 210
Stirling’s formula 150
Sturm’s Theorem 16
Subtraction of complex numbers 1 3
Summation of series by residues 116
Summation of trigonometrical series 126 127
Tangent to a cubic 197
Tannery’s theorem 371
Taylor’s series 82 95
Taylor’s series, absolute convergence of 83
Taylor’s series, remainder in 296
Transformations 7
Transformations, bilinear 8 9
Transformations, geometrical representation of 8
Transformations, linear 7 8
Transformations, rational 8; see Landen Logarithmic
Trigonometrical series, summation of 126
Uniformly convergent series 92 127
Uniformly convergent series, continuity of 92
Uniformly convergent series, differentiation of 93
Uniformly convergent series, integration of 93
Uniformly convergent series, power series 95
Uniformly convergent series, Weierstrass’s  test for 94
Variable, complex 7
Variable, independent 7
Vectors 2
Weierstrass see Sigma and Zeta functions and Uniformly convergent series
Weierstrassian elliptic function 106 169 180
Weierstrassian elliptic function, addition of semi-period, or third of period, to argument 187 331
Weierstrassian elliptic function, addition theorem 185 331
Weierstrassian elliptic function, differential equation satisfied by 183
Weierstrassian elliptic function, duplication formula for 186
Weierstrassian elliptic function, elliptic functions in terms of 191
Weierstrassian elliptic function, geometric application of 196
Weierstrassian elliptic function, in terms of sigma functions 190
Weierstrassian elliptic function, invariants of 184 194
Weierstrassian elliptic function, Legendre’s relation for 188
Weierstrassian elliptic function, order of 182
Weierstrassian elliptic function, periods of 169 180 195 196
Weierstrassian elliptic function, poles of 181 182
Weierstrassian elliptic function, relation to Jacobian functions 201
Weierstrassian elliptic function, residue at pole of 181
Weierstrassian elliptic function, transition to Jacobian functions 198
Weierstrassian elliptic function, values when one period resl and one Weierstrassian elliptic function, purely imaginary 194
Weierstrassian elliptic function, zeros of first derivative of 182 184
Weierstrassian elliptic integral 167 185 195 196
Weierstrassian elliptic integral, inversion of 169 185
Weierstrass’s Theorem 108
Well-poised series 366
Well-poised series, terminating 368 369
Whipple’s theorem for terminating well-poised series 369
Whittaker’s functions 351
Whittaker’s functions, asymptotic expansion of 352
zeros 1 39 67 118 119
Zeros, of order 39 83
Zeros, simple 39
Zeta functions, Weierstrass’s 106
Zeta functions, Weierstrass’s, elliptic functions in terms of 188
Zeta functions, Weierstrass’s, properties of 187
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