We can see from figure 4 that except , all the other Bessel functions go to zero as the argument goes to zero. Only approaches as its argument approaches zero. All Bessel functions have oscillatory behavior and their amplitude slowly decreases as the argument increases.
Figure 5 shows the behavior of the Neumann function as a function of its argument, . The important thing to note is, the Bessel functions are finite for all values of the argument, whereas the Neumann functions are finite for all values of argument except zero. When the argument tends to zero, the Neumann functions tend to .

These are called the Modified Bessel functions of first and second kind respectively.

Note: Since is imaginary, is a real quantity. So the argument of the modified Bessel functions is real.