Index P

$\wp$ -function, see Weierstrass elliptic functions.
packing analysis
- incomplete beta functions §8.24(ii)
Padé approximations §3.11(iv)—§3.11(iv)
- computation of coefficients §3.11(iv)
- convergence §3.11(iv)
- Padé table §3.11(iv)
Painlevé equations §32.2(i), see also Painlevé transcendents.
- affine Weyl groups §32.7(viii)
- alternative forms §32.2(iii)
- Bäcklund transformations §32.7—§32.7(viii)
- coalescence cascade §32.2(vi)
- compatibility conditions §32.4(i)—§32.4(v)
- elementary solutions §32.8—§32.9(iii)
- elliptic form §32.2(iv)
- graphs of solutions §32.3—§32.3(iii)
- Hamiltonian structure §32.6—§32.6(v)
- interrelations §32.7—§32.7(vii)
- isomonodromy problems §32.4(i)
  - compatibility condition §32.4(i)
- Lax pair §32.4(i)
- rational solutions §32.8—§32.8(vi)
- renormalizations §32.2(ii)
- special function solutions §32.10—§32.10(vi)
  - Airy functions §32.10(ii)
  - Bessel functions §32.10(iii)
  - Hermite polynomials §32.10(iv)
  - hypergeometric function §15.17(i), §32.10(vi)
  - parabolic cylinder functions §32.10(iv)
  - Whittaker functions §32.10(v)
- symmetric forms §32.2(v)
Painlevé property §32.2(i)
- applications §32.16
Painlevé transcendents §32.2(i), see also Painlevé equations.
- applications Ch.32
  - Boussinesq equation §32.13(iii)
  - combinatorics §32.14
  - enumerative topology §32.16
  - integrable continuous dynamical systems §32.16
  - integral equations §32.5
  - Ising model §32.16
  - Korteweg–de Vries equation §32.13(i)
  - modified Korteweg–de Vries equation §32.13(i)
  - orthogonal polynomials §32.15—§32.15
  - partial differential equations §32.13—§32.13(iii)
  - quantum gravity §32.16
  - sine-Gordon equation §32.13(ii)
  - statistical physics §32.16
  - string theory §32.16
- asymptotic approximations §32.11—§32.12(iii)
  - complex variables §32.12
  - real variables §32.11—§32.11(v)
- Bäcklund transformations §32.7—§32.7(viii)
- computation §32.17
- differential equations for §32.2(i)
- graphs §32.3—§32.3(iii)
- Hamiltonians §32.6—§32.6(v)
- Lax pair §32.4(i)
- notation §32.2(i), §32.7—§32.7(viii)
parabolic cylinder functions §12.1, §12.14
- addition theorems §12.13(i)
- applications
  - mathematical §12.16—§12.16
  - physical §12.17
- approximations §12.20
- asymptotic expansions for large parameter, see uniform asymptotic expansions for large parameter.
- asymptotic expansions for large variable §12.14(viii), §12.9
  - exponentially-improved §12.16, §12.9(ii)
- computation §12.18
- connection formulas §12.14(iv), §12.2(v)
- continued fraction §12.6
- definitions §12.14, §12.2(i), §12.2(vi)
- derivatives §12.8(ii)
- differential equations §12.2
  - numerically satisfactory solutions §12.14(i), §12.2(i)
  - standard solutions §12.2
- envelope functions §14.15(v)
- expansions in Chebyshev series §12.20
- generalized §12.15
- graphics
  - complex variables §12.3(ii)
  - real variables §12.14(iii), §12.3(i)—§12.3(i)
- Hermite polynomial case §12.1, §12.7(i)
- integral representations
  - along the real line §12.14(vi), §12.5(i)
  - compendia §12.5(iv)
  - contour integrals §12.5(ii)
  - Mellin–Barnes type §12.5(iii)
- integral transforms §12.16
- integrals §12.12—§12.12
  - asymptotic methods §12.16
  - compendia §12.12
  - Nicholson-type §12.12
- modulus and phase functions §12.14(x), §12.2(vi)
- notation §12.1
- orthogonality §12.16
- power-series expansions §12.14(v), §12.4
- recurrence relations §12.8(i)
- reflection formulas §12.2(iv)
- relations to other functions
  - Bessel functions §10.16, §12.14(vii)
  - confluent hypergeometric functions §12.14(vii), §12.7(iv), §13.18(iv), §13.6(iv)
  - error and related functions §12.7(ii)
  - Hermite polynomials §12.7(i)
  - modified Bessel functions §10.39, §12.7(iii)
  - probability functions §12.7(ii)
  - repeated integrals of the complementary error function §7.18(iv)
- sums §12.13
- tables §12.19
- uniform asymptotic expansions for large parameter §12.10—§12.10(viii), §12.14(ix)—§12.14(ix)
  - double asymptotic property §12.10(vi)
  - in terms of Airy functions §12.10(vii)—§12.10(viii), §12.14(ix)
  - in terms of elementary functions §12.10(ii)—§12.10(v), §12.14(ix)
  - modified expansions in terms of Airy functions §12.10(vii)
  - modified expansions in terms of elementary functions §12.10(vi)
- values at $z=0$ §12.14(ii), §12.2(ii)
- Wronskians §12.14(ii), §12.2(iii)
- zeros
  - asymptotic expansions for large parameter §12.11(iii)
  - asymptotic expansions for large variable §12.11(ii), §12.14(xi)
  - distribution §12.11(i)
paraboloidal coordinates
- wave equation §13.28(i)
- Whittaker–Hill equation §28.32(ii)
paraboloidal wave functions §28.31(iii)
- asymptotic behavior for large variable §28.31(iii)
- orthogonality properties §28.31(iii)
- reflection properties §28.31(iii)
parallelepiped
- volume §1.6(i), §1.6(ii)
parallelogram
- area §1.6(i)
parametrization of algebraic equations
- Jacobian elliptic functions §22.18(i)
parametrized surfaces
- area §1.6(v)
- integral over §1.6(v)
- of revolution §1.6(v)
- orientation §1.6(v)
- smooth §1.6(v)
- sphere §1.6(v)
- tangent vector §1.6(v)
paraxial wave equation §36.10(iv)
Parseval-type formulas
- Mellin transform §1.14(iv), §2.5(i)
Parseval’s equality
- orthonormal expansions §1.3(iv)
Parseval’s Formula
- Fourier series §1.8(i)
Parseval’s formula
- Fourier cosine and sine transforms §1.14(ii)
- Fourier transform §1.14(i)
partial derivative §1.5(i)
partial differential equations
- nonlinear
  - Weierstrass elliptic functions §23.21(ii)
- Painlevé transcendents §32.13
partial differentiation §1.5(i)
partial fractions §1.2(iii), see also infinite partial fractions.
particle scattering
- Coulomb functions §33.22(ii)
partition, see partition function.
partition function
- asymptotic expansion §27.14(iii)
- calculation §27.14(iii)
- divisibility §27.14(v)
- generating function §27.14(ii)
- hadronic matter §5.20
- parts §27.14(i)
- Ramanujan congruences §27.14(v)
- unrestricted §27.14(i)
partitional shifted factorial §35.4(i)
partitions §26.12(iv), §26.2—§26.4(i), §26.8—§35.4(i)
- applications §26.19, §26.20
- compositions §26.11
- conjugate §26.9(i)
- definition §26.2
- of a set §26.2—§26.4(i), §26.8(i)—§26.8(vii)
- of integers §26.10(vi), §26.2, §26.9—§26.9
- parts §26.2
- plane, see plane partitions.
- restricted, see restricted integer partitions.
- tables Table 26.12.1, Table 26.2.1, §26.21
- weight of §35.4(i)
path
- integrals of vector-valued functions §1.6(iv)
  - length §1.6(iv)
PCFs, see parabolic cylinder functions.
Pearcey integral §36.2(ii)
- asymptotic approximations §36.11—§36.12(iii)
- convergent series §36.8
- definition §36.2(ii)
- differential equation §36.10(ii)
- formula for Stokes set §36.5(ii)
- integral identities §36.9
- picture of Stokes set §36.5(iv)
- pictures of modulus Figure 36.3.1, Figure 36.3.1, Figure 36.3.1
- pictures of phase Figure 36.3.13, Figure 36.3.13
- scaling laws §36.6
- zeros §36.7(ii)
  - table Table 36.7.1
pendulum
- amplitude ( $\operatorname{am}$ ) function §22.19(i)
- Jacobian elliptic functions §22.19(i)
- Mathieu functions §28.33(iii)
pentagonal numbers
- number theory §27.14(ii)
periodic Bernoulli functions §24.2(iii)
periodic Euler functions §24.2(iii)
periodic zeta function
- relation to Hurwitz zeta function §25.13
- relation to polylogarithms §25.13
permutations §26.13—§26.16, §26.2
- adjacent transposition §26.13
- cycle notation §26.13
- definition §26.2
- derangement §26.13
- derangement number §26.13
- descent §26.14(i)
- even or odd §26.13
- excedance §26.14(i)
  - weak §26.14(i)
- fixed points §26.13
- generating function §26.14(ii)
- greater index §26.14(i)
- identities §26.14(iii)
- inversion numbers §26.13, §26.14(i), §26.15, §26.16
- major index §26.14(i), §26.16
- matrix notation §26.15
- multiset §26.16
- order notation §26.14
- restricted position §26.15
- sign §26.13, §26.15
- special values §26.14(iv)
- transpositions §26.13
- twelvefold way §26.17
Pfaff–Saalschütz formula
- ${{}_{3}F_{2}}$ functions of matrix argument §35.8(iii)
phase principle §1.10(iv), §3.8(v)
photon scattering
- hypergeometric function §15.18
pi
- computation to high precision via elliptic integrals §19.35(i)
Picard–Fuchs equations
- generalized hypergeometric functions §16.23(i)

Picard’s theorem §1.10(iii)
piecewise continuous §1.4(ii)
piecewise differentiable curve §1.6(iv)
pion-nucleon scattering
- Coulomb functions §33.22(iv)
pionic atoms
- Coulomb functions §33.22(iv)
Planck’s radiation function §4.44
plane algebraic curves, see algebraic curves.
plane curves
- elliptic integrals §19.30—§19.30(iii)
- Jacobian elliptic functions §22.18(i)
plane partitions
- applications §26.19, §26.20
- complementary §26.12(i)
- definitions §26.12(i)
- descending §26.12(i)
- generating functions §26.12(ii)
- limiting form §26.12(iv)
- recurrence relation §26.12(iii)
- strict shifted §26.12(i)
- symmetric §26.12(i)
- table Table 26.12.1
plane polar coordinates, see polar coordinates.
plasma dispersion function §7.21
plasma waves
- error functions §7.21
plasmas
- hypergeometric function §15.18
Pochhammer double-loop contour Figure 13.4.1, Figure 13.4.1, Figure 13.4.1, Figure 15.6.1, Figure 15.6.1, Figure 15.6.1, §31.9(i)
Pochhammer’s integral
- beta function §5.12
- Heun functions §31.9(i)
Pochhammer’s symbol §5.2(iii)
point sets in complex plane
- closed §1.9(ii)
- closure §1.9(ii)
- compact §1.9(vii)
- connected §1.9(ii)
- domain §1.9(ii)
  - exterior §1.9(iii)
  - interior §1.9(iii)
- open §1.9(ii)
- region §1.9(ii)
points in complex plane
- accumulation §1.9(ii)
- at infinity §1.9(iv)
- boundary §1.9(ii)
- interior §1.9(ii)
- limit (or limiting) §1.9(ii)
Poisson identity
- discrete analog §20.11(i)
  - Gauss sum §20.11(i)
Poisson integral §1.15(v), §1.9(iii)
- conjugate §1.15(v)
- harmonic functions §1.9(iii)
Poisson kernel §1.15(iii), §18.2(xii)
- Fourier integral §1.15(v)
- Fourier series §1.15(iii)
Poisson’s equation
- in channel-like geometries §14.31(i)
Poisson’s integral
- Bessel functions §10.9(i)
Poisson’s summation formula
- Fourier series §1.8(iv)—§1.8(iv)
polar coordinates §1.5(ii)
polar representation
- complex numbers §1.9(i)
pole §1.10(iii)
- movable §32.2(i)
- multiplicity §1.10(iii)
- order §1.10(iii)
Pollaczek polynomials §18.35
- of type 1 §18.35(i)
- of type 2 §18.35(i)
  - generating function §18.35(iii)
  - not in Szegő class §18.35(ii)
  - orthogonality properties §18.35(ii), §18.35(ii)
  - relation to hypergeometric function §18.35(i)
  - relations to other orthogonal polynomials §18.35(iii)
- of type 3 §18.35(i)
  - as associated type 2 Pollaczek polynomials §18.35(i)
  - orthogonality properties §18.35(ii)
  - recurrence relation §18.35(i)
polygamma functions §5.15
- computation §5.21
- continued fractions §5.15
- definition §5.15
- properties §5.15
- special values §5.15
- sums §5.16
- tables §5.22(ii)
polylogarithms §25.12(ii)
- analytic properties §25.12(ii)
- computation §25.18(i)
- definitions §25.12(ii)
- integral representations §25.12(ii)
- relations to other functions
  - Fermi–Dirac integrals §25.12(iii)
  - Lerch’s transcendent §25.14(i)
  - periodic zeta function §25.13
  - Riemann zeta function §25.12(ii)
- series expansions §25.12(ii)
- tables 3rd item
polynomials
- characteristic §3.2(iv)
- deflation §3.8(iv)
- discriminant §1.11(ii)
- monic §1.11(ii), §3.5(v)
- nodal §3.3(i)
- stable, see stable polynomials.
- Wilkinson’s §3.8(vi)
- zeros, see zeros of polynomials.
- zonal, see zonal polynomials.
polynomials orthogonal on the unit circle §18.33—§18.33
- alternative set-up §18.33(vi)
- biorthogonal §18.33(v)
- connection with orthogonal polynomials on the line §18.33(iii)
- definition §18.33(i)
- recurrence relations §18.33(ii), §18.33(vi)
- special cases §18.33(iv)
population biology
- incomplete gamma functions §8.24(i)
poristic polygon constructions of Poncelet
- Jacobian elliptic functions §22.8(iii)
positive definite
- Taylor series §1.5(iii)
potential theory
- conical functions §14.31(ii)
- symmetric elliptic integrals §19.18(ii), §19.33(ii)
power function §4.2(iv)
- analytic properties §4.2(iv)
- branch cut Figure 4.2.1, Figure 4.2.1, Figure 4.2.1
- definition §4.2(iv)
- derivatives §4.7(ii)
- general bases §4.2(iv)
- general value §4.2(iv)
- identities §4.8(ii)
- limits §4.4(iii)
- modulus §4.2(iv)
- notation §4.2(iv)
- phase §4.2(iv)
- principal value §4.2(iv)
- special values §4.4(ii)
power series
- addition §1.9(vi)
- convergence §1.9(vi)
  - circle of §1.9(vi)
  - radius of §1.9(vi)
- differentiation §1.9(vi)
- multiplication §1.9(vi)
- of logarithms §1.9(vi)
- of powers §1.9(vi)
- of reciprocals §1.9(vi)
- subtraction §1.9(vi)
primality testing
- Weierstrass elliptic functions §23.20(iii)
prime number theorem §27.11, §27.12, §27.2(i)
- equivalent statement §25.16(i)
prime numbers
- applications §27.16
- asymptotic formula §27.12
- computation Ch.27—§27.21
- counting §27.18
- cryptography §27.16
- distribution §25.16(i), §27.2(i)
  - asymptotic estimate §27.2(i)
- Euler–Fermat theorem §27.2(i)
- in arithmetic progressions
  - Dirichlet’s theorem §25.15(ii), §27.11
- Jacobi symbol §27.9
- largest known §27.12
- Legendre symbol §27.9
- Mersenne prime §27.12, §27.18
- prime number theorem §27.11, §27.12, §27.2(i)
- quadratic reciprocity law §27.9
- relation to logarithmic integral §6.16(ii)
- tables §27.2(ii), §27.21
primes, see prime numbers.
primitive Dirichlet characters
- relation to generalized Bernoulli polynomials §24.16(ii)
principal branches, see principal values.
principal values §4.2(i), see also Cauchy principal values.
- closed definition §4.2(i)
principle of the argument, see phase principle.
Pringsheim’s theorem for continued fractions §1.12(v)
probability distribution
- symmetric elliptic integrals §19.31
probability functions §12.7(ii), §7.1, §7.18(iv)
- Gaussian §7.1
- normal §7.1
- relations to other functions
  - error functions §7.1
  - parabolic cylinder functions §12.7(ii)
  - repeated integrals of the complementary error function §12.7(ii), §7.18(iv)
problème des ménages §26.15
projective coordinates §23.20(ii)
projective quantum numbers
- $\mathit{3j}$ symbols §34.2
prolate spheroidal coordinates §30.13(i)
- Laplacian §30.13(iii)
- metric coefficients §30.13(ii)
Prym’s functions §8.1
pseudo-spectral methods
- numerical solution of differential equations §18.38(i)
pseudo-spectral representations
- via classical orthogonal polynomials §18.38(i)
- via non-classical orthogonal polynomials §18.38(i)
pseudo-spectral theory and representations
- applications
  - quadrature §18.38(i)
pseudoperiodic solutions
- of Hill’s equation §28.29(ii)
- of Mathieu’s equation §28.12(ii), §28.2(iv)
pseudoprime test §27.12
pseudorandom numbers §27.19
psi function §5.2(i)
- analytic properties §5.2(i)
- applications
  - mathematical §5.19(i)
- approximations
  - Chebyshev series §5.23(ii)
  - complex variable §5.23(iii)
  - rational §5.23(i), §5.23(iii)
- asymptotic expansion §5.11(i)
- computation §5.21
- continued fractions §5.10
- definition §5.2(i)
- expansions in partial fractions §5.7(ii)
- graphics §5.3(i), §5.3(ii)
- inequalities §5.6(i)
- integral representations §5.9(ii)
- multiplication formula §5.5(iii)
- notation §5.1
- recurrence relation §5.5(i)
- reflection formula §5.5(ii)
- relation to hypergeometric function §15.4(iii)
- special values §5.4(ii)—§5.4(ii)
- tables §5.22(ii), §5.22(iii)
- Taylor series §5.7(i)
- zeros §5.2(i)
  - asymptotic approximation §5.4(iii)
  - table of §5.4(iii)
public key codes §27.16
punctured neighborhood §1.10(iii)

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