Bessel Function Calculator: J, Y, I and K Functions

Your details

J and Y are standard Bessel functions. I and K are modified (hyperbolic) Bessel functions.
Integer order from 0 to 5. Order 0 and 1 are the most common in physics and engineering.
The real argument x. Y and K are undefined at x = 0 or x < 0.
Function value
0.22389078

Value of the selected Bessel function at the given order and argument

Evaluated expressionJ_0(2)
Asymptotic approximation0.19673862
Derivative at x-0.57672481
Wronskian check0.31830989
Function value0.22389078
Asymptotic approx0.19673862
Derivative-0.57672481

Bessel J (first kind): J_0(2) = 0.223891

  • The value of J_0(2) is 0.223891.
  • The asymptotic approximation (0.196739) has a 12.1% relative error here - asymptotic forms are only accurate when |x| is much larger than the order.
  • J_n(x) oscillates for large x with slowly decaying amplitude proportional to 1/sqrt(x). These arise in problems with cylindrical or circular symmetry such as drum vibrations and heat conduction in cylinders.

Next stepTo explore how the function behaves across a range, use the chart below. To check your result, the Wronskian W[J_n, Y_n] = 2/(pi*x) should hold for any x > 0.

 = Powered by OnlyCalculators