Relationship between the Bessel function and Modified Bessel function

Question

I am confused with the relationship between the Bessel function, $J_\nu$, and the Modified Bessel function, $I_\nu$. I am finding in literature that when you have a general solution

$y(x)=c_1J_\nu(ix)+c_2J_{-\nu}(ix) \hspace{30 mm}(1)$

it can become

$y(x)=c_1I_\nu(x)+c_2I_{-\nu}(x) \hspace{33 mm}(2)$

However, I am also finding that

$J_\nu(ix)=i^\nu I_\nu(x) $ and $J_{-\nu}(ix)=i^{-\nu} I_{-\nu}(x) $

so shouldn't

$y(x)=c_1J_\nu(ix)+c_2J_{-\nu}(ix) \hspace{10 mm} \longrightarrow \hspace{10 mm} y(x)=c_1i^\nu I_\nu(x)+c_2i^{-\nu} I_{-\nu}(x)$

The only example I have seen of $(1)\rightarrow (2)$ is when $\nu = 0$. Is the relationship between $(1)$ and $(2)$ only true when $\nu = 0$? If $\nu = \frac{2}{3}$ shouldn't

$y(x)=c_1J_\frac{2}{3}(ix)+c_2J_{-\frac{2}{3}}(ix) \hspace{10 mm} \longrightarrow \hspace{10 mm}y(x)=c_1i^\frac{2}{3} I_\frac{2}{3}(x)+c_2i^{-\frac{2}{3}} I_{-\frac{2}{3}}(x)$

Any help would be appreciated. Thanks.

Answer 1

The best solution I have found is from the text "Extended Surface Heat Transfer" by Allan D. Kraus in Appendix A.

"Recognizing $J_\nu(ix)$ to be a solution..., and seeking only particular solutions of the Bessel equation, it is certain that a solution multiplied by a constant is still a solution. The solution ... may therefore also be written ... "

$y(x)=c_1I_\nu(x)+c_2I_{−\nu}(x)$

Therefore, $i^\nu$ is a constant and is essentially absorbed into the constants, $c_1$ and $c_2$.