第二类修饰贝塞尔函数的导数是否有特定的R函数？

Question

My problem is as it says in the title, I am trying to use the derivative (with respect to v ) of the modified Bessel function of the second kind K_v(x) but with no success.

I read in one of the documentation that besselDK(v,x) would work as a derivative, apparently this is not a recognized function in R. I tried to use the expansion for the derivative, namely

besselK(v,x)*(1- (1/2v) -log(e*x/2v))

but this doesn't work to give me the correct plot as well. I am trying to plot a function which includes this.

P <- function(x) (1/2)*log(exp(1)/(2*pi*x^(2)))+(3*exp(1/x^(2))/(sqrt(2*pi*x^(2))))*besselK((1/x^(2)),1/2)*(log(exp(1)/x^(2)))
x <- seq(0.1,2,0.01)
plot(x, P(x), xlim=c(0,2), ylim=c(0,1.2), type="l")

From the code above, I get a straight line as a plot. In the correct plot, it should be a curve bending between 1 and 1.5, could someone please tell me the right way to go about it?

Answer 1

The derivative at nu = 1/2 is given here .

f <- function(nu,x){
  besselK(x, nu)
}

library(gsl) # for expint_E1
fprime <- function(x){
  sqrt(pi/2/x) * expint_E1(2*x) * exp(x)
}

nu <- 1/2
h <- 1e-6
x <- 2
(f(nu+h, x) - f(nu,x)) / h 
## [1] 0.02474864
fprime(x)
## [1] 0.02474864