Some mathematics programs
The compiled programs
The uncompiled programs + compiler
PrimeDecomp: The prime factors and their multiplicity of a natural number.
Sylvester: Sylvester's formula for the number of ways into which an even number can be written as a sum of two primes.
Fourier: The Fourier-Mellin transform of sin(x) and sin(x) recreated from this function.
Moebius: Sinus curve recreated as a sum of saw-tooth functions being smaller and smaller.
Merten: The sum ∑n = 1, ..., N-1μ(n).
NumbPrimes: The number of primes less than N: 1. the exact number, 2. Gram's formula, 3. the Li series, 4. N/log(N).
Riemann's zeta function: ζ(z) = ∑n = 1, ..., ∞1/nz:
ZetaFunc: The graph of Riemann's zeta function along a line parallel with the y-axis.
ZeroFunc: The graph of the function y → │ζ(0.5 + iy)│ which determines the non-trivial zeros of Riemann's zeta function.
Newton: Newton-iteration for Riemann's zeta function.
Riemann: The graph of Riemann's formula for "the number of primes less than a given number" in which all the trivial and the first n non-trivial zeros are included.
Zerolines: The zerolines for the real and imaginary part of Riemann's zeta function.
MacZeta: The value of Riemann's zeta function calculated by the Euler-Maclaurin formula.
PrimeZeta: The zeta function Z(z) = ∑p prime 1/pz. The graph and the integral formula (containing Z(z)) for the number of primes less than N.
Vinogradov: Vinogradov's formula for the ∑p odd prime < Nzp (N large). The value for z being a rational point on the unit circle.
Recursion: The Bernoulli numbers calculated by recursion.
PrimeSums: The integral formula for the average of the sums of the prime factors for the first N natural numbers.
The units used by the programs:
compnumb: the four basic arithmetical operations for complex numbers plus arg(z), xz, log(z), sin(z) and cos(z).
functions: calculation of LiC(x) and LiR(xz) (integral logarithm), θ(x) and H(x) (Riemann), prime(n), μ(n) (Möbius) and φ(n).
bernou: the first 16 Bernoulli numbers.
gammazeta: calculation of γ(z), γ'(z), ζ(z) and ζ'(z).
zeropoint: the ordinates of the first 79 zeros for ζ(z) on the line Re(z) = 0.5.
Tomography (the roentgen scanning principle)
Scan: Scans a picture in BMP format called "pict.bmp". Enter the number of angles (e.g. 600) and the number of displacements (e.g. 600). The result is a picture called "trans_pict.bmp".
Recreate: Recreates "pict.bmp" from "trans_pict.bmp". Enter the width and the height of the wanted picture (and a light-factor, e.g. 1) - the result is called "pict0.bmp".
Planet movement round a double star
PlanetCases: Three cases.
Planet: The movement for a given start position.
Noergaard: The infinity series of Per Nørgård.
PN_generel: A generalization of the Nørgård construction.
Roel: the Roel series (wavelength parts of the Nørgård series).
KAaR: the Karl-Aage Rasmussen series.
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