# Boost Math - Special Functions

## Special Math Functions

The Special Functions section of the Boost Math library cover a broad range of areas

### Number Series

    bernoulli_b2n(10)
    #> [1] -529.1242
    max_bernoulli_b2n()
    #> [1] 129
    unchecked_bernoulli_b2n(10)
    #> [1] -529.1242
    bernoulli_b2n(start_index = 0, number_of_bernoullis_b2n = 10)
    #>  [1]  1.00000000  0.16666667 -0.03333333  0.02380952 -0.03333333  0.07575758
    #>  [7] -0.25311355  1.16666667 -7.09215686 54.97117794
    tangent_t2n(10)
    #> [1] 2.908889e+13
    tangent_t2n(start_index = 0, number_of_tangent_t2n = 10)
    #>  [1]            0            1            2           16          272
    #>  [6]         7936       353792     22368256   1903757312 209865342976
    prime(10)
    #> [1] 31
    max_prime()
    #> [1] 9999
    fibonacci(10)
    #> [1] 55
    unchecked_fibonacci(10)
    #> [1] 55

### Gamma Functions

    # Gamma function for z = 5
    tgamma(5)
    #> [1] 24
    # Gamma function for (1 + z) - 1, where z = 5
    tgamma1pm1(5)
    #> [1] 119
    # Logarithm of the gamma function for z = 5
    lgamma_boost(5)
    #> [1] 3.178054
    # Digamma function for z = 5
    digamma_boost(5)
    #> [1] 1.506118
    # Trigamma function for z = 5
    trigamma_boost(5)
    #> [1] 0.221323
    # Polygamma function of order 1 for z = 5
    polygamma(1, 5)
    #> [1] 0.221323
    # Ratio of gamma functions for a = 5, b = 3
    tgamma_ratio(5, 3)
    #> [1] 12
    # Ratio of gamma functions with delta for a = 5, delta = 2
    tgamma_delta_ratio(5, 2)
    #> [1] 0.03333333
    # Normalised lower incomplete gamma function P(a, z) for a = 5, z = 2
    gamma_p(5, 2)
    #> [1] 0.05265302
    # Normalised upper incomplete gamma function Q(a, z) for a = 5, z = 2
    gamma_q(5, 2)
    #> [1] 0.947347
    # Full lower incomplete gamma function for a = 5, z = 2
    tgamma_lower(5, 2)
    #> [1] 1.263672
    # Full upper incomplete gamma function for a = 5, z = 2
    tgamma_upper(5, 2)
    #> [1] 22.73633
    # Inverse of the normalised upper incomplete gamma function for a = 5, q = 0.5
    gamma_q_inv(5, 0.5)
    #> [1] 4.670909
    # Inverse of the normalised lower incomplete gamma function for a = 5, p = 0.5
    gamma_p_inv(5, 0.5)
    #> [1] 4.670909
    # Inverse of the normalised upper incomplete gamma function with respect to a for z = 2, q = 0.5
    gamma_q_inva(2, 0.5)
    #> [1] 2.323489
    # Inverse of the normalised lower incomplete gamma function with respect to a for z = 2, p = 0.5
    gamma_p_inva(2, 0.5)
    #> [1] 2.323489
    # Derivative of the normalised upper incomplete gamma function for a = 5, z = 2
    gamma_p_derivative(5, 2)
    #> [1] 0.09022352

### Factorials and Binomial Coefficients

    # Factorial of 5
    factorial_boost(5)
    #> [1] 120
    # Unchecked factorial of 5 (using table lookup)
    unchecked_factorial(5)
    #> [1] 120
    # Maximum factorial value that can be computed
    max_factorial()
    #> [1] 170
    # Double factorial of 6
    double_factorial(6)
    #> [1] 48
    # Rising factorial of 3 with exponent 2
    rising_factorial(3, 2)
    #> [1] 12
    # Falling factorial of 3 with exponent 2
    falling_factorial(3, 2)
    #> [1] 6
    # Binomial coefficient "5 choose 2"
    binomial_coefficient(5, 2)
    #> [1] 10

### Beta Functions

    # Euler beta function B(2, 3)
    beta_boost(2, 3)
    #> [1] 0.08333333
    # Normalised incomplete beta function I_x(2, 3) for x = 0.5
    ibeta(2, 3, 0.5)
    #> [1] 0.6875
    # Normalised complement of the incomplete beta function 1 - I_x(2, 3) for x = 0.5
    ibetac(2, 3, 0.5)
    #> [1] 0.3125
    # Full incomplete beta function B_x(2, 3) for x = 0.5
    beta_boost(2, 3, 0.5)
    #> [1] 0.05729167
    # Full complement of the incomplete beta function 1 - B_x(2, 3) for x = 0.5
    betac(2, 3, 0.5)
    #> [1] 0.02604167
    # Inverse of the normalised incomplete beta function I_x(2, 3) = 0.5
    ibeta_inv(2, 3, 0.5)
    #> [1] 0.3857276
    # Inverse of the normalised complement of the incomplete beta function I_x(2, 3) = 0.5
    ibetac_inv(2, 3, 0.5)
    #> [1] 0.3857276
    # Inverse of the normalised complement of the incomplete beta function I_x(a, b)
    # with respect to a for x = 0.5 and q = 0.5
    ibetac_inva(3, 0.5, 0.5)
    #> [1] 3
    # Inverse of the normalised incomplete beta function I_x(a, b)
    # with respect to b for x = 0.5 and p = 0.5
    ibeta_invb(0.8, 0.5, 0.5)
    #> [1] 0.8
    # Inverse of the normalised complement of the incomplete beta function I_x(a, b)
    # with respect to b for x = 0.5 and q = 0.5
    ibetac_invb(2, 0.5, 0.5)
    #> [1] 2
    # Derivative of the incomplete beta function with respect to x for a = 2, b = 3, x = 0.5
    ibeta_derivative(2, 3, 0.5)
    #> [1] 1.5

### Error Functions

    # Error function
    erf(0.5)
    #> [1] 0.5204999
    # Complementary error function
    erfc(0.5)
    #> [1] 0.4795001
    # Inverse error function
    erf_inv(0.5)
    #> [1] 0.4769363
    # Inverse complementary error function
    erfc_inv(0.5)
    #> [1] 0.4769363

### Polynomials

    # Legendre polynomial of the first kind P_2(0.5)
    legendre_p(2, 0.5)
    #> [1] -0.125
    # Derivative of the Legendre polynomial of the first kind P_2'(0.5)
    legendre_p_prime(2, 0.5)
    #> [1] 1.5
    # Zeros of the Legendre polynomial of the first kind P_2
    legendre_p_zeros(2)
    #> [1] 0.5773503
    # Legendre polynomial of the first kind with order 1 P_2^1(0.5)
    legendre_p_m(2, 1, 0.5)
    #> [1] -1.299038
    # Legendre polynomial of the second kind Q_2(0.5)
    legendre_q(2, 0.5)
    #> [1] -0.8186633
    # Next Legendre polynomial of the first kind P_3(0.5) using P_2(0.5) and P_1(0.5)
    legendre_next(2, 0.5, legendre_p(2, 0.5), legendre_p(1, 0.5))
    #> [1] -0.4375
    # Next Legendre polynomial of the first kind with order 1 P_3^1(0.5) using P_2^1(0.5) and P_1^1(0.5)
    legendre_next_m(2, 1, 0.5, legendre_p_m(2, 1, 0.5), legendre_p_m(1, 1, 0.5))
    #> [1] -0.3247595
    # Laguerre polynomial of the first kind L_2(0.5)
    laguerre(2, 0.5)
    #> [1] 0.125
    # Laguerre polynomial of the first kind with order 1 L_2^1(0.5)
    laguerre_m(2, 1, 0.5)
    #> [1] 1.625
    # Next Laguerre polynomial of the first kind L_3(0.5) using L_2(0.5) and L_1(0.5)
    laguerre_next(2, 0.5, laguerre(2, 0.5), laguerre(1, 0.5))
    #> [1] -0.1458333
    # Next Laguerre polynomial of the first kind with order 1 L_3^1(0.5) using L_2^1(0.5) and L_1^1(0.5)
    laguerre_next_m(2, 1, 0.5, laguerre_m(2, 1, 0.5), laguerre_m(1, 1, 0.5))
    #> [1] 1.479167
    # Hermite polynomial H_2(0.5)
    hermite(2, 0.5)
    #> [1] -1
    # Next Hermite polynomial H_3(0.5) using H_2(0.5) and H_1(0.5)
    hermite_next(2, 0.5, hermite(2, 0.5), hermite(1, 0.5))
    #> [1] -5
    # Chebyshev polynomial of the first kind T_2(0.5)
    chebyshev_t(2, 0.5)
    #> [1] -0.5
    # Chebyshev polynomial of the second kind U_2(0.5)
    chebyshev_u(2, 0.5)
    #> [1] 0
    # Derivative of the Chebyshev polynomial of the first kind T_2'(0.5)
    chebyshev_t_prime(2, 0.5)
    #> [1] 2
    # Next Chebyshev polynomial of the first kind T_3(0.5) using T_2(0.5) and T_1(0.5)
    chebyshev_next(0.5, chebyshev_t(2, 0.5), chebyshev_t(1, 0.5))
    #> [1] -1
    # Chebyshev polynomial of the first kind using Clenshaw's recurrence with coefficients
    # c = c(1, 0, -1) at x = 0.5
    chebyshev_clenshaw_recurrence(c(1, 0, -1), 0.5)
    #> [1] 1
    # Chebyshev polynomial of the first kind using Clenshaw's recurrence with interval [0, 1]
    chebyshev_clenshaw_recurrence_ab(c(1, 0, -1), 0, 1, 0.5)
    #> [1] 1.5
    # Spherical harmonic function Y_2^1(0.5, 0.5)
    spherical_harmonic(2, 1, 0.5, 0.5)
    #> [1] -0.2852481-0.1558318i
    # Real part of the spherical harmonic function Y_2^1(0.5, 0.5)
    spherical_harmonic_r(2, 1, 0.5, 0.5)
    #> [1] -0.2852481
    # Imaginary part of the spherical harmonic function Y_2^1(0.5, 0.5)
    spherical_harmonic_i(2, 1, 0.5, 0.5)
    #> [1] -0.1558318
    # Gegenbauer polynomial C_2^(1)(0.5)
    gegenbauer(2, 1, 0.5)
    #> [1] 0
    # Derivative of the Gegenbauer polynomial C_2^(1)'(0.5)
    gegenbauer_prime(2, 1, 0.5)
    #> [1] 4
    # k-th derivative of the Gegenbauer polynomial C_2^(1)''(0.5)
    gegenbauer_derivative(2, 1, 0.5, 2)
    #> [1] 8
    # Jacobi polynomial P_2^(1, 2)(0.5)
    jacobi(2, 1, 2, 0.5)
    #> [1] -0.1875
    # Derivative of the Jacobi polynomial P_2^(1, 2)'(0.5)
    jacobi_prime(2, 1, 2, 0.5)
    #> [1] 3.75
    # Second derivative of the Jacobi polynomial P_2^(1, 2)''(0.5)
    jacobi_double_prime(2, 1, 2, 0.5)
    #> [1] 10.5
    # 3rd derivative of the Jacobi polynomial P_2^(1, 2)^(k)(0.5)
    jacobi_derivative(2, 1, 2, 0.5, 3)
    #> [1] 0

### Bessel Functions

    # Bessel function of the first kind J_0(1)
    cyl_bessel_j(0, 1)
    #> [1] 0.7651977
    # Bessel function of the second kind Y_0(1)
    cyl_neumann(0, 1)
    #> [1] 0.08825696
    # Modified Bessel function of the first kind I_0(1)
    cyl_bessel_i(0, 1)
    #> [1] 1.266066
    # Modified Bessel function of the second kind K_0(1)
    cyl_bessel_k(0, 1)
    #> [1] 0.4210244
    # Spherical Bessel function of the first kind j_0(1)
    sph_bessel(0, 1)
    #> [1] 0.841471
    # Spherical Bessel function of the second kind y_0(1)
    sph_neumann(0, 1)
    #> [1] -0.5403023
    # Derivative of the Bessel function of the first kind J_0(1)
    cyl_bessel_j_prime(0, 1)
    #> [1] -0.4400506
    # Derivative of the Bessel function of the second kind Y_0(1)
    cyl_neumann_prime(0, 1)
    #> [1] 0.7812128
    # Derivative of the modified Bessel function of the first kind I_0(1)
    cyl_bessel_i_prime(0, 1)
    #> [1] 0.5651591
    # Derivative of the modified Bessel function of the second kind K_0(1)
    cyl_bessel_k_prime(0, 1)
    #> [1] -0.6019072
    # Derivative of the spherical Bessel function of the first kind j_0(1)
    sph_bessel_prime(0, 1)
    #> [1] -0.3011687
    # Derivative of the spherical Bessel function of the second kind y_0(1)
    sph_neumann_prime(0, 1)
    #> [1] 1.381773
    # Finding the first zero of the Bessel function of the first kind J_0
    cyl_bessel_j_zero(0, 1)
    #> [1] 2.404826
    # Finding the first zero of the Bessel function of the second kind Y_0
    cyl_neumann_zero(0, 1)
    #> [1] 0.893577
    # Finding multiple zeros of the Bessel function of the first kind J_0 starting from index 1
    cyl_bessel_j_zero(0, start_index = 1, number_of_zeros = 5)
    #> [1]  2.404826  5.520078  8.653728 11.791534 14.930918
    # Finding multiple zeros of the Bessel function of the second kind Y_0 starting from index 1
    cyl_neumann_zero(0, start_index = 1, number_of_zeros = 5)
    #> [1]  0.893577  3.957678  7.086051 10.222345 13.361097

### Hankel Functions

    cyl_hankel_1(2, 0.5)
    #> [1] 0.03060402-5.441371i
    cyl_hankel_2(2, 0.5)
    #> [1] 0.03060402+5.441371i
    sph_hankel_1(2, 0.5)
    #> [1] 0.01637111-25.05992i
    sph_hankel_2(2, 0.5)
    #> [1] 0.01637111+25.05992i

### Airy Functions

    airy_ai(2)
    #> [1] 0.03492413
    airy_bi(2)
    #> [1] 3.298095
    airy_ai_prime(2)
    #> [1] -0.05309038
    airy_bi_prime(2)
    #> [1] 4.100682
    airy_ai_zero(1)
    #> [1] -2.338107
    airy_bi_zero(1)
    #> [1] -1.173713
    airy_ai_zero(start_index = 1, number_of_zeros = 5)
    #> [1] -2.338107 -4.087949 -5.520560 -6.786708 -7.944134
    airy_bi_zero(start_index = 1, number_of_zeros = 5)
    #> [1] -1.173713 -3.271093 -4.830738 -6.169852 -7.376762

### Elliptic Integrals

    # Carlson's elliptic integral Rf with parameters x = 1, y = 2, z = 3
    ellint_rf(1, 2, 3)
    #> [1] 0.7269459
    # Carlson's elliptic integral Rd with parameters x = 1, y = 2, z = 3
    ellint_rd(1, 2, 3)
    #> [1] 0.2904603
    # Carlson's elliptic integral Rj with parameters x = 1, y = 2, z = 3, p = 4
    ellint_rj(1, 2, 3, 4)
    #> [1] 0.2398481
    # Carlson's elliptic integral Rc with parameters x = 1, y = 2
    ellint_rc(1, 2)
    #> [1] 0.7853982
    # Carlson's elliptic integral Rg with parameters x = 1, y = 2, z = 3
    ellint_rg(1, 2, 3)
    #> [1] 1.401847
    # Incomplete elliptic integral of the first kind with k = 0.5, phi = pi/4
    ellint_1(0.5, pi / 4)
    #> [1] 0.8043661
    # Complete elliptic integral of the first kind
    ellint_1(0.5)
    #> [1] 1.68575
    # Incomplete elliptic integral of the second kind with k = 0.5, phi = pi/4
    ellint_2(0.5, pi / 4)
    #> [1] 0.767196
    # Complete elliptic integral of the second kind
    ellint_2(0.5)
    #> [1] 1.467462
    # Incomplete elliptic integral of the third kind with k = 0.5, n = 0.5, phi = pi/4
    ellint_3(0.5, 0.5, pi / 4)
    #> [1] 0.8930657
    # Complete elliptic integral of the third kind with k = 0.5, n = 0.5
    ellint_3(0.5, 0.5)
    #> [1] 2.413672
    # Incomplete elliptic integral D with k = 0.5, phi = pi/4
    ellint_d(0.5, pi / 4)
    #> [1] 0.1486805
    # Complete elliptic integral D
    ellint_d(0.5)
    #> [1] 0.8731526
    # Jacobi zeta function with k = 0.5, phi = pi/4
    jacobi_zeta(0.5, pi / 4)
    #> [1] 0.06698741
    # Heuman's lambda function with k = 0.5, phi = pi/4
    heuman_lambda(0.5, pi / 4)
    #> [1] 0.6632254

### Jacobi Elliptic Functions

    # Jacobi Elliptic Functions
    k <- 0.5
    u <- 2
    jacobi_elliptic(k, u)
    #> $sn
    #> [1] 0.9628982
    #>
    #> $cn
    #> [1] -0.269865
    #>
    #> $dn
    #> [1] 0.8764741
    # Individual Jacobi Elliptic Functions
    jacobi_cd(k, u)
    #> [1] -0.3078984
    jacobi_cn(k, u)
    #> [1] -0.269865
    jacobi_cs(k, u)
    #> [1] -0.2802632
    jacobi_dc(k, u)
    #> [1] -3.247825
    jacobi_dn(k, u)
    #> [1] 0.8764741
    jacobi_ds(k, u)
    #> [1] 0.9102458
    jacobi_nc(k, u)
    #> [1] -3.705557
    jacobi_nd(k, u)
    #> [1] 1.140935
    jacobi_ns(k, u)
    #> [1] 1.038531
    jacobi_sc(k, u)
    #> [1] -3.568074
    jacobi_sd(k, u)
    #> [1] 1.098604
    jacobi_sn(k, u)
    #> [1] 0.9628982

### Jacobi Theta Functions

    # Jacobi Theta Functions
    x <- 0.5
    q <- 0.9
    tau <- 1.5
    jacobi_theta1(x, q)
    #> [1] 0.0001025529
    jacobi_theta1tau(x, tau)
    #> [1] 0.2951461
    jacobi_theta2(x, q)
    #> [1] 0.5090257
    jacobi_theta2tau(x, tau)
    #> [1] 0.5403556
    jacobi_theta3(x, q)
    #> [1] 0.5090257
    jacobi_theta3tau(x, tau)
    #> [1] 1.009707
    jacobi_theta3m1(x, q)
    #> [1] -0.4909743
    jacobi_theta3m1tau(x, tau)
    #> [1] 0.00970738
    jacobi_theta4(x, q)
    #> [1] 0.0001025529
    jacobi_theta4tau(x, tau)
    #> [1] 0.9902926
    jacobi_theta4m1(x, q)
    #> [1] -0.9998974
    jacobi_theta4m1tau(x, tau)
    #> [1] -0.009707391

### Lambert W Function

    # Lambert W Function (Principal Branch)
    lambert_w0(0.3)
    #> [1] 0.2367553
    # Lambert W Function (Branch -1)
    lambert_wm1(-0.3)
    #> [1] -1.781337
    # Derivative of the Lambert W Function (Principal Branch)
    lambert_w0_prime(0.3)
    #> [1] 0.6381087
    # Derivative of the Lambert W Function (Branch -1)
    lambert_wm1_prime(-0.3)
    #> [1] -7.599525

### Zeta Functions

    # Riemann Zeta Function
    zeta(2)  # Should return pi^2 / 6
    #> [1] 1.644934

### Exponential Integrals

    # Exponential integral En with n = 1 and z = 2
    expint_en(1, 2)
    #> [1] 0.04890051
    # Exponential integral Ei with z = 2
    expint_ei(2)
    #> [1] 4.954234

### Hypergeometric Functions

    # Hypergeometric Function 1F0
    hypergeometric_1F0(2, 0.2)
    #> [1] 1.5625
    # Hypergeometric Function 0F1
    hypergeometric_0F1(1, 0.8)
    #> [1] 1.974957
    # Hypergeometric Function 2F0
    hypergeometric_2F0(0.1, -1, 0.1)
    #> [1] 0.99
    # Hypergeometric Function 1F1
    hypergeometric_1F1(2, 3, 1)
    #> [1] 2
    # Regularised Hypergeometric Function 1F1
    hypergeometric_1F1_regularized(2, 3, 1)
    #> [1] 1
    # Logarithm of the Hypergeometric Function 1F1
    log_hypergeometric_1F1(2, 3, 1)
    #> [1] 0.6931472
    #> attr(,"sign")
    #> [1] 1
    # Hypergeometric Function pFq
    hypergeometric_pFq(c(2, 3), c(4, 5), 6)
    #> [1] 11.32427

### Basic Functions

    # sin(pi * 0.5)
    sin_pi(0.5)
    #> [1] 1
    # cos(pi * 0.5)
    cos_pi(0.5)
    #> [1] 0
    # log(1 + 0.5)
    log1p_boost(0.5)
    #> [1] 0.4054651
    # exp(0.5) - 1
    expm1_boost(0.5)
    #> [1] 0.6487213
    cbrt(8)
    #> [1] 2
    # sqrt(1 + 0.5) - 1
    sqrt1pm1(0.5)
    #> [1] 0.2247449
    # 2^3 - 1
    powm1(2, 3)
    #> [1] 7
    hypot(3, 4)
    #> [1] 5
    rsqrt(4)
    #> [1] 0.5

### Sinus Cardinal and Hyperbolic Sinus Cardinal Functions

    # Sinus cardinal function
    sinc_pi(0.5)
    #> [1] 0.9588511
    # Hyperbolic sinus cardinal function
    sinhc_pi(0.5)
    #> [1] 1.042191

### Inverse Hyperbolic Functions

    # Inverse Hyperbolic Cosine
    acosh_boost(2)
    #> [1] 1.316958
    # Inverse Hyperbolic Sine
    asinh_boost(1)
    #> [1] 0.8813736
    # Inverse Hyperbolic Tangent
    atanh_boost(0.5)
    #> [1] 0.5493061

### Owen’s T Function

    # Owens T Function
    owens_t(1, 0.5)
    #> [1] 0.04306469