When $z$ is replaced by $\pm \mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation:

with its algebraic form

Assume first that $\nu$ is real, $q$ is positive, and $a={\lambda }_{\nu }\left(q\right)$; see §28.12(i). Write

Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. It follows that (28.20.1) has independent and unique solutions ${\mathrm{M}}_{\nu }^{(3)}(z,h)$, ${\mathrm{M}}_{\nu }^{(4)}(z,h)$ such that

as $\mathrm{\Re }z\to +\mathrm{\infty }$ with $-\pi +\delta \le \mathrm{\Im }z\le 2\pi -\delta$, and

as $\mathrm{\Re }z\to +\mathrm{\infty }$ with $-2\pi +\delta \le \mathrm{\Im }z\le \pi -\delta$. See §10.2(ii) for the notation. In addition, there are unique solutions ${\mathrm{M}}_{\nu }^{(1)}(z,h)$, ${\mathrm{M}}_{\nu }^{(2)}(z,h)$ that are real when $z$ is real and have the properties

as $\mathrm{\Re }z\to +\mathrm{\infty }$ with $|\mathrm{\Im }z|\le \pi -\delta$.

For other values of $z$, $h$, and $\nu$ the functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$, $j=1,2,3,4$, are determined by analytic continuation. Furthermore,

For $j=1,2,3,4$,

For $n=0,1,2,\mathrm{\dots }$,

For $s\in \mathbb{Z}$,

When $\nu$ is an integer the right-hand sides of (28.20.25) are replaced by the their limiting values. And for the corresponding identities for the radial functions use (28.20.15) and (28.20.16).