Hankel Function Calculator

Compute the Hankel functions H^(1)ₙ(x)=Jₙ+iYₙ and H^(2)ₙ(x)=Jₙ−iYₙ as complex numbers from order n and x, with magnitude, argument and the Bessel values Jₙ, Yₙ.

Input

Enter the order n (a non-negative integer) and a positive argument x to evaluate the first- and second-kind Hankel functions as complex numbers.

Order n

Non-negative integer

Argument x

Positive real number

Result

First-kind Hankel function H^(1) of order 0, x = 3

-0.2600519549 + 0.37685001 i

Second kind H^(2), order 0, x = 3

-0.2600519549 − 0.37685001 i

Magnitude |H^(1)|

0.4578678295

Argument arg H^(1) (radians)

2.1748250513

Bessel function J of order 0

-0.2600519549

Neumann function Y of order 0

0.37685001

Graph of J and Y for order 0

J order 0

Y order 0

How it works

The Hankel functions are complex combinations of the Bessel functions of the first and second kind: H^(1)ₙ(x)=Jₙ(x)+iYₙ(x) and H^(2)ₙ(x)=Jₙ(x)−iYₙ(x).
The headline result is the first-kind Hankel function H^(1)ₙ(x). Its real part equals the Bessel function Jₙ(x) and its imaginary part equals the Neumann function Yₙ(x).
H^(2)ₙ(x) is the complex conjugate of H^(1)ₙ(x): it shares the same real part Jₙ(x) and has the opposite-signed imaginary part −Yₙ(x).
The magnitude is sqrt(Jₙ(x)^2 + Yₙ(x)^2) and the argument is atan2(Yₙ(x), Jₙ(x)) in radians.
Yₙ(x) diverges at x=0, so only positive values of x are accepted. The order n must be a non-negative integer.
Jₙ is computed from a Maclaurin series for small x and an asymptotic expansion for large x, with a numerically stable backward recurrence (Miller method) for higher orders. Yₙ uses series representations and forward recurrence.
Results are approximations from established numerical methods; small rounding errors may appear in the last displayed digits.

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