Index T

tangent function, see trigonometric functions.
tangent numbers §24.15(ii)
- tables Table 24.15.1
Taylor series §1.10(i)
- asymptotic approximations for coefficients §2.10(iv)
Taylor’s theorem
- one variable §1.10(i), §1.4(vi)
- two variables §1.5(iii)
tempered distributions §1.16(v), §1.16(vi), §2.6(ii)
- convergence §1.16(v)
- Fourier transform §1.16(vii)
term-by-term integration §1.9(vii)
terminant function §2.11(v)
- incomplete gamma functions §8.22(i)
tesseral harmonics §14.30(i)
test functions
- distributions §1.16(i)
Theorem of Ince
- Mathieu’s equation §28.2(iv), §28.5(i)
theta functions §20.2(i)
- addition formulas §20.7(ii)
- applications
  - mathematical §20.12—§20.12(ii)
  - physical §20.13
- combined §20.11(v)
- computation §20.14
- derivatives §20.4—§20.5(ii)
  - of ratios §20.7(vii)
- discrete analog §20.11(i)
- double products §20.5(iii)
- duplication formula §20.7(iii)
- Fourier series §20.2(i)
- fundamental parallelogram §20.2(ii)
- generalizations §20.11
- graphics
  - complex variables §20.3(ii)—§20.3(iii)
  - real variables §20.3(i)—§20.3(i)
- infinite products §20.5—§20.5(iii)
- integrals §20.10
- Jacobi’s identity §20.4(i)
- Jacobi’s inversion formula §20.11(iii), §20.9(ii)
- Jacobi’s original notation §20.1
- Jacobi’s triple product §20.5(i)
- Landen transformation §20.7(vi)
- Laplace transform with respect to lattice parameter §20.10(ii)
- lattice parameter §20.1
  - transformation of §20.7(viii)
- lattice points §20.2(ii)
- limit forms as $\Im\tau\rightarrow 0+$ §20.13
- McKean and Moll’s §20.1
- Mellin transform with respect to lattice parameter §20.10(i)
- modular transformations §20.7(viii)
- multidimensional, see Chapter 21.
- Neville’s §20.1, §22.2
- nome §20.1
  - rectangular case §20.1
  - transformation of §20.7(iv), §20.7(ix)
- notation §20.1
- periodicity §20.2(ii)
- permutation symmetry §20.11(v)
- power series §20.6
- quasi-periodicity §20.2(ii)
- Ramanujan’s §20.11(ii)
- Ramanujan’s change of base §20.11(iii)
- rectangular case §20.1
- relations to other functions
  - Dedekind’s eta function Figure 20.3.2, Figure 20.3.2, Figure 20.3.2
  - elliptic integrals §20.9(i)
  - elliptic modular function §20.9(ii)
  - Jacobi’s epsilon function §22.16(ii)
  - Jacobian elliptic functions §20.9(ii), §22.2
  - modular functions §23.15—§23.15(ii)
  - Riemann zeta function §20.9(iii)
  - symmetric elliptic integrals §19.25(iv)
  - Weierstrass elliptic functions §20.9(ii), §23.6(i)
- Riemann §21.2(i)
- Riemann with characteristics §21.2(ii)
- sums of squares §20.7(i)
- tables §20.15
- translation by half-periods §20.2(iii)
- values at $z=0$ §20.4—§20.4(ii)
- Watson’s expansions §20.8
- Watson’s identities §20.7(v)
- with characteristics §20.11(iv)
- zeros §20.2(iv)
Thomae transformation
- ${{}_{3}F_{2}}$ functions of matrix argument §35.8(iii)
$3j$ symbols
- relations to other functions
  - dual Hahn polynomials §18.38(iii)
  - Hahn polynomials §18.38(iii)
$\mathit{3j},\mathit{6j},\mathit{9j}$ symbols
- relation to generalized hypergeometric functions §16.24(iii), §16.4(iii), §16.4(iii)
$\mathit{3j}$ symbols §34.2
- angular momenta §34.2
- applications §34.12
- approximations for large parameters §34.8
- computation §34.13
- definition §34.2
- Gaunt coefficient §34.3(vii)
- Gaunt’s integral §34.3(vii)
- generating functions §34.3(v)
- graphical method §34.9
- notation §34.1
- orthogonality §34.3(iv)
- projective quantum numbers §34.2
- recursion relations §34.3(iii)
- Regge symmetries §34.3(ii)
- relations to other functions
  - Legendre functions §34.3(vii)
  - rotation matrices §34.3(vii)
  - spherical harmonics §34.3(vii)
- representation as
  - finite sum of algebraic quantities §34.2
  - generalized hypergeometric functions §34.2
- special cases §34.3(i)
- summation convention §34.3(iv)
- sums §34.3(vi)
- symmetry §34.3(ii)
- tables §34.14
- triangle conditions §34.2
- zeros §34.10

Toda equation
- Hermite polynomials §18.38(ii)
tomography
- confluent hypergeometric functions §13.28(iii)
tops
- Jacobian elliptic, or hyperelliptic, integrals §22.19(iv)
toroidal coordinates §14.19(i), §14.31(i)
toroidal functions §14.19(i)
- applications §14.31(i)
- definitions §14.19(i)
- hypergeometric representations §14.19(ii)
- integral representations §14.19(iii)
- sums §14.19(iv)
- Whipple’s formula §14.19(v)
torus
- complex §20.12(ii)
transcendental equations
- asymptotic solutions §2.2
transcendental functions §32.2(i)
transition points §2.8(i), §2.9(iii)
transport equilibrium
- generalized exponential integral §8.24(iii)
triangle conditions
- $\mathit{3j}$ symbols §34.2
triangle inequality §1.9(i)
triangles
- solution of §4.42
triangular matrices
- confluent hypergeometric functions §13.27—§13.27
triconfluent Heun equation §31.12
trigonometric functions Ch.4
- addition formulas §4.21(i)
- analytic properties §4.14
- applications
  - cubic equations §4.43
  - solution of triangles and spherical triangles §4.42
- approximations §4.47
- Chebyshev-series expansions §4.47(i)
- computation §4.45(i)
- conformal maps Figure 4.15.7, Figure 4.15.7, Figure 4.15.7
- continued fractions §4.25
- definitions §4.14
- derivatives §4.20
- differential equations §4.20
- elementary properties §4.16—§4.16
- graphics
  - complex argument §4.15(iii)—§4.15(iii)
  - real argument §4.15(i)
- identities §4.21
- inequalities §4.18
- infinite products §4.22
- integrals
  - definite §4.26(iii)
  - indefinite §4.26(ii)
- inverse, see inverse trigonometric functions.
- Laurent series §4.19
- limits §4.17
- Maclaurin series §4.19
- moduli §4.21(iv)
- multiples of argument §4.21(iii)
- notation §4.1
- orthogonality §4.26(iii)
- partial fractions §4.22
- periodicity §4.14
- poles §4.28
- real and imaginary parts §4.21(iv)
- relations to hyperbolic functions §4.28
- special values Table 4.17.1
- squares and products §4.21(ii)
- sums §4.27
- tables §4.46
- zeros §4.14
triple integrals §1.5(v)
truncated exponential series §8.11(v)
turning points §2.8(i), §2.9(iii)
- fractional or multiple §2.8(v)
two-body relativistic scattering
- Lamé polynomials §29.19(ii)

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