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The  $  2 \pi $-
periodic solutions of the [[Mathieu equation|Mathieu equation]]

$$ 

\frac{d  ^ {2} u }{dz  ^ {2} }
 + ( a + 16q  \cos  2z) u  =  0,\ \ 
z \in \mathbf R ,
$$

which exist only when the point  $  ( a, q) $
in the parameter plane lies on the boundary of the stability zones. A Mathieu function is even or odd, and is unique up to a factor; the second linearly-independent solution grows linearly in  $  z $
for  $  | z | \rightarrow \infty $, 
provided  $  q \neq 0 $. 
The even Mathieu functions are the eigen functions of the integral equation

$$ 
G( z)  =  \lambda \int\limits _ {- \pi } ^  \pi   e ^ {k  \cos  z  \cos  t } G( t)  dt,\ \ 
k  =  \sqrt 32q .
$$

An analogous equation is satisfied by the odd Mathieu functions. The notation for Mathieu functions is:

$$ 
ce _ {0} ( z, q), ce _ {1} ( z, q) , .  .  . ; \  se _ {1} ( z, q) , se _ {2} ( z, q) , .  .  . .
$$

For  $  q \rightarrow 0 $
these functions reduce to the [[Trigonometric system|trigonometric system]]

$$ 
1 , \cos  z , .  .  . ; \ \ 
\sin  z, \sin  2z \dots
$$

and they possess the same orthogonality properties on the interval  $  (- \pi , \pi ) $. 
The Mathieu functions admit Fourier-series expansions which converge for small  $  | q | \leq  r _ {n} $; 
the coefficients of these series are convergent power series in  $  q $, 
for example,

$$ 
ce _ {0} ( z, q)  = \ 
1 +
$$

$$ 
+ 
\sum _{n=1} ^  \infty   \left [ 2  ^ {n+1}
\frac{q  ^ {n} }{( n!)
 ^ {2} }
 - 
\frac{n( 3n+ 4) 2  ^ {n+3} q  ^ {n+2} }{(( n+ 1)!)
 ^ {2} }
 + O( q  ^ {n+4} ) \right ]  \cos  2nz.
$$

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 2</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''3. Automorphic functions''' , McGraw-Hill  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Sansone,   "Equazioni differenziali nel campo reale" , '''1''' , Zanichelli  (1948)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.J.O. Strett,   "Lamésche-, Mathieusche- und verwandte Funktionen in Physik und Technik" , Springer  (1932)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.W. Mac-Lachlan,   "Theory and application of Mathieu functions" , Clarendon Press  (1947)</TD></TR>
</table>