By setting $$f = \frac{y}{\sqrt{z}}$$, transform bessel's equation of order m, $$z^{2}\frac{d^{2}f}{dz^{2}} + z\frac{df}{dz} + (z^{2}-m^{2})f = 0 $$ into $$\frac{d^{2}y}{dz^{2}} + y(1 + \frac{1}{4z^{2}})- \frac{m^{2}}{z^{2}}$$
Assuming the transformation has been done so.
The second and third part of the problem is what I do not understand.
b) Use this equation to determine the exact expressions for $$J_{\frac{1}{2}} \text and Y_{\frac{1}{2}}$$. Each expression should contain 2 unknown coefficients.
c) use the asymptotic form of $$J_{m}(z) \text and Y_{m}(z)$$ close to z=0 to determine three of the 4 unknown coefficients.
Please help me. Really struggling big time with this question for a while.
There is a typo in the wording. You wrote :
The correct result of the transform is : $$\frac{d^{2}y}{dz^{2}} + y(1 + \frac{1}{4z^{2}})- \frac{m^{2}}{z^{2}}y = 0$$ Then, in case of $m=\frac{1}{2}$, the solution of the Bessel ODE is $$f(z)=c_1 J_{1/2}(z) +c_2 Y_{1/2}(z)$$ and after transformation, the ODE becomes $$\frac{d^{2}y}{dz^{2}} + y=0$$ You can solve it in terms of circular functions (again with two constants). Then, relate it with $y(z)=f(z)\sqrt{z}$
The asymptotic form at $z=0$ allows to distinguish $J_{1/2}(z)$ and $Y_{1/2}(z)$ because the first is finite and the second infinite. So, the expressions of $J_{1/2}(z)$ and $Y_{1/2}(z)$ can be obtained separately.