How this calculator works
Bessel functions occur in radial wave, heat and vibration models. This tool sums the defining first-kind series, using the complex principal logarithm for non-integer powers. Negative integer orders use a symmetry identity. The modest argument limit reduces cancellation in the series; it is not a limit on the mathematical function. Values are approximate, especially near a zero or a negative integer order.
Formula
J_v(z) = (z/2)^v sum from k=0 to infinity of (-z^2/4)^k / (k! Gamma(v+k+1)). J_(-n)(z) = (-1)^n J_n(z) for integer n.
Worked example
At z = 0, J_0(0) = 1. For z = i and order 0, every series term is positive: 1 + 1/4 + 1/64 + ... = 1.2660658778. Its imaginary part is zero.
Advertisement
Frequently asked questions
Which Bessel function is evaluated?
The first kind J only. Second-kind Y, modified K and Hankel functions are different functions.
Can a real input produce a complex answer?
Yes. A negative real argument with non-integer order uses a principal complex power.
Why reject arguments above 12 in magnitude?
Large alternating terms can cancel and lose digits. A specialised asymptotic or recurrence library is preferable beyond this supported series range.