---
title: "Boost Math - Special Functions"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Boost Math - Special Functions}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup, include=FALSE}
library(boostmath)
```

## Special Math Functions

The [Special Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/special.html) section of the Boost Math library cover a broad range of areas

### [Number Series](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/number_series.html)

```{r}
bernoulli_b2n(10)
max_bernoulli_b2n()
unchecked_bernoulli_b2n(10)
bernoulli_b2n(start_index = 0, number_of_bernoullis_b2n = 10)
tangent_t2n(10)
tangent_t2n(start_index = 0, number_of_tangent_t2n = 10)
prime(10)
max_prime()
fibonacci(10)
unchecked_fibonacci(10)
```

### [Gamma Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/sf_gamma.html)

```{r}
# Gamma function for z = 5
tgamma(5)
# Gamma function for (1 + z) - 1, where z = 5
tgamma1pm1(5)
# Logarithm of the gamma function for z = 5
lgamma_boost(5)
# Digamma function for z = 5
digamma_boost(5)
# Trigamma function for z = 5
trigamma_boost(5)
# Polygamma function of order 1 for z = 5
polygamma(1, 5)
# Ratio of gamma functions for a = 5, b = 3
tgamma_ratio(5, 3)
# Ratio of gamma functions with delta for a = 5, delta = 2
tgamma_delta_ratio(5, 2)
# Normalised lower incomplete gamma function P(a, z) for a = 5, z = 2
gamma_p(5, 2)
# Normalised upper incomplete gamma function Q(a, z) for a = 5, z = 2
gamma_q(5, 2)
# Full lower incomplete gamma function for a = 5, z = 2
tgamma_lower(5, 2)
# Full upper incomplete gamma function for a = 5, z = 2
tgamma_upper(5, 2)
# Inverse of the normalised upper incomplete gamma function for a = 5, q = 0.5
gamma_q_inv(5, 0.5)
# Inverse of the normalised lower incomplete gamma function for a = 5, p = 0.5
gamma_p_inv(5, 0.5)
# Inverse of the normalised upper incomplete gamma function with respect to a for z = 2, q = 0.5
gamma_q_inva(2, 0.5)
# Inverse of the normalised lower incomplete gamma function with respect to a for z = 2, p = 0.5
gamma_p_inva(2, 0.5)
# Derivative of the normalised upper incomplete gamma function for a = 5, z = 2
gamma_p_derivative(5, 2)
```

### [Factorials and Binomial Coefficients](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/factorials.html)

```{r}
# Factorial of 5
factorial_boost(5)
# Unchecked factorial of 5 (using table lookup)
unchecked_factorial(5)
# Maximum factorial value that can be computed
max_factorial()
# Double factorial of 6
double_factorial(6)
# Rising factorial of 3 with exponent 2
rising_factorial(3, 2)
# Falling factorial of 3 with exponent 2
falling_factorial(3, 2)
# Binomial coefficient "5 choose 2"
binomial_coefficient(5, 2)
```

### [Beta Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/sf_beta.html)

```{r}
# Euler beta function B(2, 3)
beta_boost(2, 3)
# Normalised incomplete beta function I_x(2, 3) for x = 0.5
ibeta(2, 3, 0.5)
# Normalised complement of the incomplete beta function 1 - I_x(2, 3) for x = 0.5
ibetac(2, 3, 0.5)
# Full incomplete beta function B_x(2, 3) for x = 0.5
beta_boost(2, 3, 0.5)
# Full complement of the incomplete beta function 1 - B_x(2, 3) for x = 0.5
betac(2, 3, 0.5)
# Inverse of the normalised incomplete beta function I_x(2, 3) = 0.5
ibeta_inv(2, 3, 0.5)
# Inverse of the normalised complement of the incomplete beta function I_x(2, 3) = 0.5
ibetac_inv(2, 3, 0.5)
# 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)
# 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)
# 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)
# Derivative of the incomplete beta function with respect to x for a = 2, b = 3, x = 0.5
ibeta_derivative(2, 3, 0.5)
```

### [Error Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/sf_erf.html)

```{r}
# Error function
erf(0.5)
# Complementary error function
erfc(0.5)
# Inverse error function
erf_inv(0.5)
# Inverse complementary error function
erfc_inv(0.5)
```

### [Polynomials](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/sf_poly.html)

```{r}
# Legendre polynomial of the first kind P_2(0.5)
legendre_p(2, 0.5)
# Derivative of the Legendre polynomial of the first kind P_2'(0.5)
legendre_p_prime(2, 0.5)
# Zeros of the Legendre polynomial of the first kind P_2
legendre_p_zeros(2)
# Legendre polynomial of the first kind with order 1 P_2^1(0.5)
legendre_p_m(2, 1, 0.5)
# Legendre polynomial of the second kind Q_2(0.5)
legendre_q(2, 0.5)
# 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))
# 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))
# Laguerre polynomial of the first kind L_2(0.5)
laguerre(2, 0.5)
# Laguerre polynomial of the first kind with order 1 L_2^1(0.5)
laguerre_m(2, 1, 0.5)
# 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))
# 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))
# Hermite polynomial H_2(0.5)
hermite(2, 0.5)
# 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))
# Chebyshev polynomial of the first kind T_2(0.5)
chebyshev_t(2, 0.5)
# Chebyshev polynomial of the second kind U_2(0.5)
chebyshev_u(2, 0.5)
# Derivative of the Chebyshev polynomial of the first kind T_2'(0.5)
chebyshev_t_prime(2, 0.5)
# 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))
# 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)
# 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)
# Spherical harmonic function Y_2^1(0.5, 0.5)
spherical_harmonic(2, 1, 0.5, 0.5)
# Real part of the spherical harmonic function Y_2^1(0.5, 0.5)
spherical_harmonic_r(2, 1, 0.5, 0.5)
# Imaginary part of the spherical harmonic function Y_2^1(0.5, 0.5)
spherical_harmonic_i(2, 1, 0.5, 0.5)
# Gegenbauer polynomial C_2^(1)(0.5)
gegenbauer(2, 1, 0.5)
# Derivative of the Gegenbauer polynomial C_2^(1)'(0.5)
gegenbauer_prime(2, 1, 0.5)
# k-th derivative of the Gegenbauer polynomial C_2^(1)''(0.5)
gegenbauer_derivative(2, 1, 0.5, 2)
# Jacobi polynomial P_2^(1, 2)(0.5)
jacobi(2, 1, 2, 0.5)
# Derivative of the Jacobi polynomial P_2^(1, 2)'(0.5)
jacobi_prime(2, 1, 2, 0.5)
# Second derivative of the Jacobi polynomial P_2^(1, 2)''(0.5)
jacobi_double_prime(2, 1, 2, 0.5)
# 3rd derivative of the Jacobi polynomial P_2^(1, 2)^(k)(0.5)
jacobi_derivative(2, 1, 2, 0.5, 3)
```

### [Bessel Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/bessel.html)
```{r}
# Bessel function of the first kind J_0(1)
cyl_bessel_j(0, 1)
# Bessel function of the second kind Y_0(1)
cyl_neumann(0, 1)
# Modified Bessel function of the first kind I_0(1)
cyl_bessel_i(0, 1)
# Modified Bessel function of the second kind K_0(1)
cyl_bessel_k(0, 1)
# Spherical Bessel function of the first kind j_0(1)
sph_bessel(0, 1)
# Spherical Bessel function of the second kind y_0(1)
sph_neumann(0, 1)
# Derivative of the Bessel function of the first kind J_0(1)
cyl_bessel_j_prime(0, 1)
# Derivative of the Bessel function of the second kind Y_0(1)
cyl_neumann_prime(0, 1)
# Derivative of the modified Bessel function of the first kind I_0(1)
cyl_bessel_i_prime(0, 1)
# Derivative of the modified Bessel function of the second kind K_0(1)
cyl_bessel_k_prime(0, 1)
# Derivative of the spherical Bessel function of the first kind j_0(1)
sph_bessel_prime(0, 1)
# Derivative of the spherical Bessel function of the second kind y_0(1)
sph_neumann_prime(0, 1)
# Finding the first zero of the Bessel function of the first kind J_0
cyl_bessel_j_zero(0, 1)
# Finding the first zero of the Bessel function of the second kind Y_0
cyl_neumann_zero(0, 1)
# 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)
# 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)
```

### [Hankel Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/hankel.html)

```{r}
cyl_hankel_1(2, 0.5)
cyl_hankel_2(2, 0.5)
sph_hankel_1(2, 0.5)
sph_hankel_2(2, 0.5)
```

### [Airy Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/airy.html)

```{r}
airy_ai(2)
airy_bi(2)
airy_ai_prime(2)
airy_bi_prime(2)
airy_ai_zero(1)
airy_bi_zero(1)
airy_ai_zero(start_index = 1, number_of_zeros = 5)
airy_bi_zero(start_index = 1, number_of_zeros = 5)
```

### [Elliptic Integrals](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/ellint.html)

```{r}
# Carlson's elliptic integral Rf with parameters x = 1, y = 2, z = 3
ellint_rf(1, 2, 3)
# Carlson's elliptic integral Rd with parameters x = 1, y = 2, z = 3
ellint_rd(1, 2, 3)
# Carlson's elliptic integral Rj with parameters x = 1, y = 2, z = 3, p = 4
ellint_rj(1, 2, 3, 4)
# Carlson's elliptic integral Rc with parameters x = 1, y = 2
ellint_rc(1, 2)
# Carlson's elliptic integral Rg with parameters x = 1, y = 2, z = 3
ellint_rg(1, 2, 3)
# Incomplete elliptic integral of the first kind with k = 0.5, phi = pi/4
ellint_1(0.5, pi / 4)
# Complete elliptic integral of the first kind
ellint_1(0.5)
# Incomplete elliptic integral of the second kind with k = 0.5, phi = pi/4
ellint_2(0.5, pi / 4)
# Complete elliptic integral of the second kind
ellint_2(0.5)
# 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)
# Complete elliptic integral of the third kind with k = 0.5, n = 0.5
ellint_3(0.5, 0.5)
# Incomplete elliptic integral D with k = 0.5, phi = pi/4
ellint_d(0.5, pi / 4)
# Complete elliptic integral D
ellint_d(0.5)
# Jacobi zeta function with k = 0.5, phi = pi/4
jacobi_zeta(0.5, pi / 4)
# Heuman's lambda function with k = 0.5, phi = pi/4
heuman_lambda(0.5, pi / 4)
```

### [Jacobi Elliptic Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/jacobi.html)

```{r}
# Jacobi Elliptic Functions
k <- 0.5
u <- 2
jacobi_elliptic(k, u)
# Individual Jacobi Elliptic Functions
jacobi_cd(k, u)
jacobi_cn(k, u)
jacobi_cs(k, u)
jacobi_dc(k, u)
jacobi_dn(k, u)
jacobi_ds(k, u)
jacobi_nc(k, u)
jacobi_nd(k, u)
jacobi_ns(k, u)
jacobi_sc(k, u)
jacobi_sd(k, u)
jacobi_sn(k, u)
```

### [Jacobi Theta Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/jacobi_theta.html)

```{r}
# Jacobi Theta Functions
x <- 0.5
q <- 0.9
tau <- 1.5
jacobi_theta1(x, q)
jacobi_theta1tau(x, tau)
jacobi_theta2(x, q)
jacobi_theta2tau(x, tau)
jacobi_theta3(x, q)
jacobi_theta3tau(x, tau)
jacobi_theta3m1(x, q)
jacobi_theta3m1tau(x, tau)
jacobi_theta4(x, q)
jacobi_theta4tau(x, tau)
jacobi_theta4m1(x, q)
jacobi_theta4m1tau(x, tau)
```

### [Lambert W Function](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/lambert_w.html)

```{r}
# Lambert W Function (Principal Branch)
lambert_w0(0.3)
# Lambert W Function (Branch -1)
lambert_wm1(-0.3)
# Derivative of the Lambert W Function (Principal Branch)
lambert_w0_prime(0.3)
# Derivative of the Lambert W Function (Branch -1)
lambert_wm1_prime(-0.3)
```

### [Zeta Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/zetas.html)

```{r}
# Riemann Zeta Function
zeta(2)  # Should return pi^2 / 6
```

### [Exponential Integrals](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/expint.html)

```{r}
# Exponential integral En with n = 1 and z = 2
expint_en(1, 2)
# Exponential integral Ei with z = 2
expint_ei(2)
```

### [Hypergeometric Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/hypergeometric.html)

```{r}
# Hypergeometric Function 1F0
hypergeometric_1F0(2, 0.2)
# Hypergeometric Function 0F1
hypergeometric_0F1(1, 0.8)
# Hypergeometric Function 2F0
hypergeometric_2F0(0.1, -1, 0.1)
# Hypergeometric Function 1F1
hypergeometric_1F1(2, 3, 1)
# Regularised Hypergeometric Function 1F1
hypergeometric_1F1_regularized(2, 3, 1)
# Logarithm of the Hypergeometric Function 1F1
log_hypergeometric_1F1(2, 3, 1)
# Hypergeometric Function pFq
hypergeometric_pFq(c(2, 3), c(4, 5), 6)
```

### [Basic Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/powers.html)

```{r}
# sin(pi * 0.5)
sin_pi(0.5)
# cos(pi * 0.5)
cos_pi(0.5)
# log(1 + 0.5)
log1p_boost(0.5)
# exp(0.5) - 1
expm1_boost(0.5)
cbrt(8)
# sqrt(1 + 0.5) - 1
sqrt1pm1(0.5)
# 2^3 - 1
powm1(2, 3)
hypot(3, 4)
rsqrt(4)
```

### [Sinus Cardinal and Hyperbolic Sinus Cardinal Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/sinc.html)

```{r}
# Sinus cardinal function
sinc_pi(0.5)
# Hyperbolic sinus cardinal function
sinhc_pi(0.5)
```

### [Inverse Hyperbolic Functions](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/inv_hyper.html)

```{r}
# Inverse Hyperbolic Cosine
acosh_boost(2)
# Inverse Hyperbolic Sine
asinh_boost(1)
# Inverse Hyperbolic Tangent
atanh_boost(0.5)
```

### [Owen's T Function](https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/owens_t.html)

```{r}
# Owens T Function
owens_t(1, 0.5)
```
