About the Project
NIST

tangent numbers

AdvancedHelp

(0.001 seconds)

1—10 of 24 matching pages

1: 24.15 Related Sequences of Numbers
§24.15(ii) Tangent Numbers
24.15.4 T 2 n - 1 = ( - 1 ) n - 1 2 2 n ( 2 2 n - 1 ) 2 n B 2 n , n = 1 , 2 , ,
24.15.5 T 2 n = 0 , n = 0 , 1 , .
Table 24.15.1: Genocchi and Tangent numbers.
n 0 1 2 3 4 5 6 7 8
2: 24.19 Methods of Computation
For example, the tangent numbers T n can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied. …
3: Bibliography K
  • D. E. Knuth and T. J. Buckholtz (1967) Computation of tangent, Euler, and Bernoulli numbers. Math. Comp. 21 (100), pp. 663–688.
  • 4: 4.33 Maclaurin Series and Laurent Series
    4.33.3 tanh z = z - z 3 3 + 2 15 z 5 - 17 315 z 7 + + 2 2 n ( 2 2 n - 1 ) B 2 n ( 2 n ) ! z 2 n - 1 + , | z | < 1 2 π .
    5: 4.19 Maclaurin Series and Laurent Series
    4.19.3 tan z = z + z 3 3 + 2 15 z 5 + 17 315 z 7 + + ( - 1 ) n - 1 2 2 n ( 2 2 n - 1 ) B 2 n ( 2 n ) ! z 2 n - 1 + , | z | < 1 2 π ,
    4.19.9 ln ( tan z z ) = n = 1 ( - 1 ) n - 1 2 2 n ( 2 2 n - 1 - 1 ) B 2 n n ( 2 n ) ! z 2 n , | z | < 1 2 π .
    6: 21.1 Special Notation
    g , h

    positive integers.

    | S |

    number of elements of the set S .

    a b

    intersection index of a and b , two cycles lying on a closed surface. a b = 0 if a and b do not intersect. Otherwise a b gets an additive contribution from every intersection point. This contribution is 1 if the basis of the tangent vectors of the a and b cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is - 1 .

    Lowercase boldface letters or numbers are g -dimensional real or complex vectors, either row or column depending on the context. …
    7: 19.11 Addition Theorems
    19.11.6_5 R C ( γ - δ , γ ) = - 1 δ arctan ( δ sin θ sin ϕ sin ψ α 2 - 1 - α 2 cos θ cos ϕ cos ψ ) .
    8: 1.9 Calculus of a Complex Variable
    §1.9(i) Complex Numbers
    Polar Representation
    Modulus and Phase
    Powers
    Winding Number
    9: 3.5 Quadrature
    For the Bernoulli numbers B m see §24.2(i). … The w k are also known as Christoffel coefficients or Christoffel numbers and they are all positive. The remainder is given by …
    3.5.45 erfc λ = e - λ 2 2 π - π π e - λ 2 tan 2 ( 1 2 θ ) d θ .
    Table 3.5.20 gives the results of applying the composite trapezoidal rule (3.5.2) with step size h ; n indicates the number of function values in the rule that are larger than 10 - 15 (we exploit the fact that the integrand is even). …
    10: 19.8 Quadratic Transformations
    When a 0 and g 0 are positive numbers, define …
    ϕ 1 = ϕ + arctan ( k tan ϕ ) = arcsin ( ( 1 + k ) sin ϕ cos ϕ 1 - k 2 sin 2 ϕ ) .