Spherical Hankel Function Calculator
Enter the order n and argument x to compute the spherical Hankel functions h(1)n(x)=jn+i yn and h(2)n(x)=jn−i yn as complex numbers, along with the magnitude and the spherical Bessel components.
Input
Enter the order n and real argument x to compute the spherical Hankel functions h(1)n(x)=jn+i yn and h(2)n(x)=jn−i yn as complex numbers.
Enter a non-negative integer.
Enter a nonzero real number.
Result
h(1) at order 0, x = 3
0.0470400027 + 0.3299974989i
h(2) at order 0
0.0470400027 − 0.3299974989i
Magnitude |h(1) order 0|
0.3333333333
Component breakdown
| Quantity | Value |
|---|---|
| Spherical Bessel j order 0 | 0.0470400027 |
| Spherical Neumann y order 0 | 0.3299974989 |
| Real part of h(1) | 0.0470400027 |
| Imaginary part of h(1) | 0.3299974989 |
| Real part of h(2) | 0.0470400027 |
| Imaginary part of h(2) | -0.3299974989 |
Curves of j order 0 and y order 0
j order 0
y order 0
How it works
- The spherical Hankel functions are complex functions built from the spherical Bessel function jn(x) and the spherical Neumann function yn(x): the first kind h(1)n(x)=jn(x)+i yn(x) and the second kind h(2)n(x)=jn(x)−i yn(x).
- The spherical Bessel function jn(x) starts from j0(x)=sin(x)/x and j1(x)=sin(x)/x^2−cos(x)/x. It uses upward recurrence when x is larger than the order and the backward Miller recurrence when x is small, for numerical stability.
- The spherical Neumann function yn(x) starts from y0(x)=−cos(x)/x and y1(x)=−cos(x)/x^2−sin(x)/x and is evaluated with the always-stable upward recurrence y(n+1)=(2n+1)/x times yn minus y(n−1).
- h(1)n and h(2)n are complex conjugates of each other, so their magnitudes are equal: |h(1)n(x)|=|h(2)n(x)|=sqrt(jn(x)^2+yn(x)^2).
- Because yn(x) diverges as x approaches 0, x=0 is not allowed. The order n must be a non-negative integer.
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Spherical Hankel Function Calculator