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1: Sidebar 9.SB2: Interference Patterns in Caustics
The bright sharp-edged triangle is a caustic, that is a line of focused light. …
2: Sidebar 5.SB1: Gamma & Digamma Phase Plots
This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices. … The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.
3: Sidebar 9.SB1: Supernumerary Rainbows
The faint line below the main colored arc is a ‘supernumerary rainbow’, produced by the interference of different sun-rays traversing a raindrop and emerging in the same direction. …
4: 25.18 Methods of Computation
§25.18(ii) Zeros
Most numerical calculations of the Riemann zeta function are concerned with locating zeros of ζ ( 1 2 + i t ) in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of ζ ( s ) lie on the critical line s = 1 2 . Calculations to date (2008) have found no nontrivial zeros off the critical line. …
5: Karl Dilcher
Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. …
6: 25.10 Zeros
§25.10(i) Distribution
The Riemann hypothesis states that all nontrivial zeros lie on this line. … Calculations relating to the zeros on the critical line make use of the real-valued function … Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of ζ ( s ) in the critical strip are on the critical line (van de Lune et al. (1986)). More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …
7: 4.3 Graphics
See accompanying text
Figure 4.3.1: ln x and e x . Parallel tangent lines at ( 1 , 0 ) and ( 0 , 1 ) make evident the mirror symmetry across the line y = x , demonstrating the inverse relationship between the two functions. Magnify
Lines parallel to the real axis in the z -plane map onto rays in the w -plane, and lines parallel to the imaginary axis in the z -plane map onto circles centered at the origin in the w -plane. …
8: 7.19 Voigt Functions
7.19.4 H ( a , u ) = a π e t 2 d t ( u t ) 2 + a 2 = 1 a π 𝖴 ( u a , 1 4 a 2 ) .
H ( a , u ) is sometimes called the line broadening function; see, for example, Finn and Mugglestone (1965). …
9: 25.17 Physical Applications
Analogies exist between the distribution of the zeros of ζ ( s ) on the critical line and of semiclassical quantum eigenvalues. …
10: 26.22 Software
  • On-Line Encyclopedia of Integer Sequences (website).