Editing Picard theorem (section)

==Generalization and current research==
''Great Picard's theorem'' is true in a slightly more general form that also applies to [[meromorphic function]]s:

<blockquote> '''Great Picard's Theorem (meromorphic version):''' If ''M'' is a [[Riemann surface]], ''w'' a point on ''M'', '''P'''<sup>1</sup>('''C''') =&nbsp;'''C'''&nbsp;∪&nbsp;{∞} denotes the [[Riemann sphere]] and ''f'' : ''M''\{''w''} → '''P'''<sup>1</sup>('''C''') is a holomorphic function with essential singularity at ''w'', then on any open subset of ''M'' containing ''w'', the function ''f''(''z'') attains all but at most ''two'' points of '''P'''<sup>1</sup>('''C''') infinitely often.</blockquote>

'''Example:''' The function ''f''(''z'') =&nbsp;1/(1&nbsp;−&nbsp;''e''<sup>1/''z''</sup>) is meromorphic on '''C*''' = '''C''' - {0}, the complex plane with the origin deleted. It has an essential singularity at ''z''&nbsp;=&nbsp;0 and attains the value ∞ infinitely often in any neighborhood of 0; however it does not attain the values 0 or 1.

With this generalization, ''Little Picard Theorem'' follows from ''Great Picard Theorem'' because an entire function is either a polynomial or it has an essential singularity at infinity. As with the little theorem, the (at most two) points that are not attained are lacunary values of the function.

The following [[conjecture]] is related to "Great Picard's Theorem":<ref>{{Cite journal|last = Elsner|first = B.|year = 1999|journal = [[Annales de l'Institut Fourier]]|volume = 49|pages = 303–331|title = Hyperelliptic action integral | url=http://archive.numdam.org/ARCHIVE/AIF/AIF_1999__49_1/AIF_1999__49_1_303_0/AIF_1999__49_1_303_0.pdf |issue = 1|doi = 10.5802/aif.1675|doi-access = free}}</ref>

<blockquote>'''Conjecture:''' Let {''U''<sub>1</sub>, ..., ''U<sub>n</sub>''} be a collection of open connected subsets of '''C''' that [[Cover (topology)|cover]] the punctured [[unit disk]] '''D'''&nbsp;\&nbsp;{0}. Suppose that on each ''U<sub>j</sub>'' there is an [[Injective function|injective]] [[holomorphic function]] ''f<sub>j</sub>'', such that d''f''<sub>''j''</sub> = d''f<sub>k</sub>'' on each intersection ''U''<sub>''j''</sub>&nbsp;∩&nbsp;''U''<sub>''k''</sub>. Then the differentials glue together to a [[meromorphic function|meromorphic]] 1-[[Differential form|form]] on '''D'''.</blockquote>

It is clear that the differentials glue together to a holomorphic 1-form ''g''&nbsp;d''z'' on '''D'''&nbsp;\&nbsp;{0}. In the special case where the [[residue (complex analysis)|residue]] of ''g'' at 0 is zero the conjecture follows from the "Great Picard's Theorem".