Addditional special functions (Mpmath)

• Mpmath: Conversions of parameters of elliptic functions
  • Elliptic nome \(q\): from \(q, m, k, \tau, \bar{q}\)
    • ctx.qfrom()
  • Number-theoretic nome \(\bar q = q^2\) (qbar): from \(q, m, k, \tau, \bar{q}\)
    • ctx.qbarfrom()
  • Elliptic parameter \(m\): from \(q, m, k, \tau, \bar{q}\)
    • ctx.mfrom()
  • Elliptic modulus \(k\): from \(q, m, k, \tau, \bar{q}\)
    • ctx.kfrom()
  • Elliptic period ratio \(\tau\): from \(q, m, k, \tau, \bar{q}\)
    • ctx.taufrom()
• Related to Lerch’s phi
  • Mpmath: Bernoulli number as fraction
    • mpm.bernfrac()
  • Mpmath: Dirichlet L-Series
    • ctx.dirichlet_l()
  • Mpmath: Number of zeros of the Riemann zeta function
    • ctxflint.zeta_nzeros()
  • Mpmath: Secondary zeta function
    • ctx.secondzeta()
• Additional numbertheoretic functions
  • Mpmath: Prime counting function
    • ctx.primepi()
  • Mpmath: Mangoldt function
    • ctx.mangoldt()
  • Mpmath: Upper bound for the value of the prime counting function
    • ctx.primepi2_upper()
  • Mpmath: Lower dound for the value of the prime counting function
    • ctx.primepi2_lower()
  • Mpmath, DAMath: Riemann R function
    • ctx.riemann_r()
  • Mpmath, DAMath: Prime zeta function
    • ctx.primezeta()
  • Mpmath: Mertens constant
    • ctx.const_mertens()
  • Mpmath: Twin prime constant
    • ctx.const_twinprime()
  • Mpmath: Cyclotomic polynomial
    • ctx.cyclotomic()
  • Mpmath: Stirling number of the first kind
    • ctx.stirling1()
  • Mpmath: Stirling number of the second kind
    • ctx.stirling2()
  • Mpmath: CODE!! Polyexponential function
    • ctx.polyexp()
  • Mpmath, DAMath: Moebius function, \(\mu(n)\)
    • math53.moebius()
• Generalized hypergeometric functions
  • Generalized hypergeometric function \({}_pF_q\)
    • ctx.hyperg_pfq()
  • Generalized hypergeometric function \({}_2F_3\)
    • ctx.hyperg_2f3()
  • Generalized hypergeometric function \({}_3F_2\)
    • ctx.hyperg_3f2()
  • Generalized hypergeometric function \({}_2F_2\)
    • ctx.hyperg_2f2()
  • Generalized hypergeometric function \(\,_2F_0\)
    • ctx.hyperg_2f0()
• Appell Functions
  • Appell function \(F_1\)
    • ctx.appell_f1()
  • Appell function \(F_2\)
    • ctx.appell_f2()
  • Appell function \(F_3\)
    • ctx.appell_f3()
  • Appell function \(F_4\)
    • ctx.appell_f4()
• Q Functions
  • q-Pochhammer symbol
    • ctx.q_pochhammer()
  • q-gamma function
    • ctx.q_gamma()
  • q-factorial
    • ctx.q_factorial()
  • Hypergeometric q-series
    • ctx.q_hyperg()
• Further generalizations of gamma and hypergeometric functions
  • Limit of the product of gamma functions
    • ctx.gamma_prod()
  • Limit of a weighted combination of hypergeometric functions
    • ctx.hyperg_combination()
  • Meijer G-function
    • ctx.meijer_g()
  • Bilateral hypergeometric series
    • ctx.bilateral_hyperg()
  • Generalized 2D hypergeometric series
    • ctx.hyperg_2d()