About the Project
NIST

convex functions

AdvancedHelp

(0.001 seconds)

8 matching pages

1: 5.3 Graphics
See accompanying text
Figure 5.3.2: ln Γ ( x ) . This function is convex on ( 0 , ) ; compare §5.5(iv). Magnify
2: 1.4 Calculus of One Variable
§1.4(viii) Convex Functions
A function f ( x ) is convex on ( a , b ) if … A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.
See accompanying text
Figure 1.4.2: Convex function f ( x ) . … Magnify
3: 5.5 Functional Relations
§5.5(iv) Bohr–Mollerup Theorem
If a positive function f ( x ) on ( 0 , ) satisfies f ( x + 1 ) = x f ( x ) , f ( 1 ) = 1 , and ln f ( x ) is convex (see §1.4(viii)), then f ( x ) = Γ ( x ) .
4: Bibliography L
  • J. T. Lewis and M. E. Muldoon (1977) Monotonicity and convexity properties of zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 171–178.
  • 5: 1.7 Inequalities
    For f integrable on [ 0 , 1 ] , a < f ( x ) < b , and ϕ convex on ( a , b ) 1.4(viii)), …
    6: 5.18 q -Gamma and q -Beta Functions
    Also, ln Γ q ( x ) is convex for x > 0 , and the analog of the Bohr-Mollerup theorem (§5.5(iv)) holds. …
    7: 20.3 Graphics
    §20.3(i) θ -Functions: Real Variable and Real Nome
    §20.3(ii) θ -Functions: Complex Variable and Real Nome
    In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …
    §20.3(iii) θ -Functions: Real Variable and Complex Lattice Parameter
    In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …
    8: Bibliography M
  • I. G. Macdonald (1990) Hypergeometric Functions.
  • B. Markman (1965) Contribution no. 14. The Riemann zeta function. BIT 5, pp. 138–141.
  • F. Matta and A. Reichel (1971) Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.
  • A. Michaeli (1996) Asymptotic analysis of edge-excited currents on a convex face of a perfectly conducting wedge under overlapping penumbra region conditions. IEEE Trans. Antennas and Propagation 44 (1), pp. 97–101.
  • S. C. Milne (2002) Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Ramanujan J. 6 (1), pp. 7–149.