# §28.5 Second Solutions ,

## §28.5(i) Definitions

### ¶ Theorem of Ince (1922)

If a nontrivial solution of Mathieu’s equation with has period or , then any linearly independent solution cannot have either period.

Second solutions of (28.2.1) are given by

28.5.1

when , , and by

28.5.2

when , . For , we have

28.5.3

and

28.5.4

compare §28.2(vi). The functions , are unique.

The factors and in (28.5.1) and (28.5.2) are normalized so that

As with , , , , and . This determines the signs of and . (Other normalizations for and can be found in the literature, but most formulas—including connection formulas—are unaffected since and are invariant.)

28.5.6

As a consequence of the factor on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as on .

### ¶ Wronskians

For further information on , , and expansions of , in Fourier series or in series of , functions, see McLachlan (1947, Chapter VII) or Meixner and Schäfke (1954, §2.72).

## §28.5(ii) Graphics: Line Graphs of Second Solutions of Mathieu’s Equation

### ¶ Odd Second Solutions

 Figure 28.5.1: for and (for comparison) . Symbols: : Mathieu function, : second solution, Mathieu’s equation and : real variable Referenced by: §28.5(ii) Permalink: http://dlmf.nist.gov/28.5.F1 Encodings: pdf, png Figure 28.5.2: for and (for comparison) . Symbols: : Mathieu function, : second solution, Mathieu’s equation and : real variable Permalink: http://dlmf.nist.gov/28.5.F2 Encodings: pdf, png
 Figure 28.5.3: for and (for comparison) . Symbols: : Mathieu function, : second solution, Mathieu’s equation and : real variable Permalink: http://dlmf.nist.gov/28.5.F3 Encodings: pdf, png Figure 28.5.4: for and (for comparison) . Symbols: : Mathieu function, : second solution, Mathieu’s equation and : real variable Permalink: http://dlmf.nist.gov/28.5.F4 Encodings: pdf, png

### ¶ Even Second Solutions

 Figure 28.5.5: for and (for comparison) . Symbols: : second solution, Mathieu’s equation, : Mathieu function and : real variable Permalink: http://dlmf.nist.gov/28.5.F5 Encodings: pdf, png Figure 28.5.6: for and (for comparison) . Symbols: : second solution, Mathieu’s equation, : Mathieu function and : real variable Referenced by: §28.5(ii) Permalink: http://dlmf.nist.gov/28.5.F6 Encodings: pdf, png