Question

To show ${\mathbf{N}}_{\left(2n+1\right)}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}\mathbf{+}\mathbf{1}}{\mathbf{J}}_{-\left(2n+1\right)}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$.

Short answer

It is proved that${\mathrm{N}}_{(2\mathrm{n}+1)}\left(\mathrm{x}\right)={(-1)}^{\mathrm{n}+1}{\mathrm{J}}_{-(2\mathrm{n}+1)}\left(\mathrm{x}\right)$

Step-by-step solution

Concept of Neumann Function:

The Neumann function:

${\mathit{N}}_{\mathbf{p}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{\pi p}\mathbf{\right)}{\mathbf{J}}_{\mathbf{p}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{-}{\mathbf{J}}_{\mathbf{-}\mathbf{p}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{\left(}\mathbf{\pi p}\mathbf{\right)}}$

Calculation of the function N(n+12)(x) :

Consider the Neumann function as follows:

$$\begin{gathered} {\mathrm{N}}_{\mathrm{p}}\left(\mathrm{x}\right)=\frac{\cos \left(\mathrm{\tau \tau p}\right){\mathrm{J}}_{\mathrm{p}}\left(\mathrm{x}\right)-{\mathrm{J}}_{-\mathrm{p}}\left(\mathrm{\tau \tau p}\right)}{\sin \left(\mathrm{\tau \tau p}\right)} \\ {\mathrm{N}}_{\left(\mathrm{n}+\frac{1}{2}\right)}\left(\mathrm{x}\right)=\frac{\cos (\mathrm{n}+{\displaystyle \frac{1}{2}})\left(\mathrm{\tau \tau p}\right){\mathrm{J}}_{{}_{\left(\mathrm{n}+\frac{1}{2}\right)}}^{\left(\mathrm{x}\right)-\mathrm{J}}-\left(\mathrm{n}+{\displaystyle \frac{1}{2}}\right)}{\sin \left(\mathrm{\tau \tau p}\right)\left(\mathrm{n}+\frac{1}{2}\right)} \end{gathered}$$

$$\begin{gathered} =\frac{\cos \left(\mathrm{n\tau \tau }+{\displaystyle \frac{\mathrm{\tau \tau }}{2}}\right){\mathrm{J}}_{\left(\mathrm{n}+{\displaystyle \frac{1}{2}}\right)}^{\left(\mathrm{x}\right)-\mathrm{J}}-\left(\mathrm{n}+{\displaystyle \frac{1}{2}}\right)}{\mathrm{sin\tau \tau }\left(\mathrm{n}+\frac{1}{2}\right)} \\ =\frac{-\mathrm{J}-{\left(\mathrm{n}+\frac{1}{2}\right)}^{\left(\mathrm{x}\right)}}{{\left(-1\right)}^{\mathrm{n}}} \end{gathered}$$

Simplify further as follows:

${\mathrm{N}}_{\left(\mathrm{n}+\frac{1}{2}\right)}\left(\mathrm{x}\right)={\left(-1\right)}^{\left(\mathrm{n}+\frac{1}{2}\right)}{\mathrm{J}}_{\left(\mathrm{n}+\frac{1}{2}\right)}\left(\mathrm{x}\right)$