As in Problem 1, study the K_p (X) functions. It is interesting to note (see Problem $17.4) that K_1/2(X) is equal to the asymptotic approximation.

The Wolfram Mathematica built-in functions are used because they are very convenient. The Mathematica function Bessel J[p,x] returns the values of J_p(X).

The approximate value for large X is, $K_p\left(X\right)=\sqrt{\frac{\mathrm{\pi }}{2X}}{e}^{-X}+O\left(\frac{e^{-X}}{X}\right)$.

The plot of function K_p (x) together with its asymptotic approximate can be drawn as:

For p=1/2:

K_p (x) =K_1/2 (x)

The approximate formula for small x is¸ K_p (x) =localid="1664367633363" $$" width="9">π2NP(X)

where,$$" width="9">NP(X)=p=02πInx+O(1)p>0-¬(p)π2xp+Oxpp<1OxIn1xp=1Ox2-pp>1

The plot of the function K₀(x) together with its small 'x' approximate can be drawn as:

The plot of the functionK_1/2 (x) together with its small x approximate can be drawn as:

The plot of the function K₁ (x) together with its small x approximate can be drawn as:

The plot of the function K_3/2 (x) together with its small x approximate can be drawn as:

The plot of the function K₂ (x) together with its small x approximate can be drawn as:

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