Editing Picard theorem

{{Short description|Theorem about the range of an analytic function}}
{{for|the theorem on existence and uniqueness of solutions of differential equations|Picard–Lindelöf theorem}}
{{Complex analysis sidebar}}
In [[complex analysis]], '''Picard's great theorem''' and '''Picard's little theorem''' are related [[theorem]]s about the [[range of a function|range]] of an [[analytic function]]. They are named after [[Émile Picard]].

==The theorems==
[[Image:Cplot of exp(1z).png|right|220px|thumb|[[Domain coloring]] plot of the function exp({{frac|1|''z''}}), centered on the essential singularity at ''z''&nbsp;=&nbsp;0. The hue of a point ''z'' represents the [[argument (complex analysis)|argument]] of exp({{frac|1|''z''}}), the luminance represents its absolute value. This plot shows that arbitrarily close to the singularity, all non-zero values are attained.]]

<blockquote>'''Little Picard Theorem:''' If a [[function (mathematics)|function]] <math display="inline">f: \mathbb{C} \to\mathbb{C}</math> is [[entire function|entire]] and non-constant, then the set of values that <math display="inline">f(z)</math>  assumes is either the whole complex plane or the plane minus a single point. </blockquote>

<blockquote>'''Sketch of Proof:''' Picard's original proof was based on properties of the [[modular lambda function]], usually denoted by <math display="inline">\lambda</math>, and which performs, using modern terminology, the holomorphic [[universal covering]] of the [[twice punctured]] plane by the unit disc. This function is explicitly constructed in the theory of [[elliptic functions]]. If <math display="inline">f</math> omits two values, then the composition of <math display="inline">f</math> with the inverse of the modular function maps the plane into
the unit disc which implies that <math display="inline">f</math> is constant by [[Liouville's theorem (complex analysis)|Liouville's theorem.]]</blockquote>

This theorem is a significant strengthening of Liouville's theorem which states that the image of an entire non-constant function must be [[unbounded function|unbounded]]. Many different proofs of Picard's theorem were later found and [[Schottky's theorem]] is a quantitative version of it. In the case where the values of <math display="inline">f</math> are missing a single point, this point is called a [[lacunary value]] of the function.

<blockquote>'''Great Picard's Theorem:''' If an analytic function <math display="inline">f</math> has an [[essential singularity]] at a point <math display="inline">w</math> , then on any [[punctured neighborhood]] of <math display="inline">w, f(z)</math>  takes on all possible complex values, with at most a single exception, infinitely often.</blockquote>

This is a substantial strengthening of the [[Casorati–Weierstrass theorem]], which only guarantees that the range of <math display="inline">f</math>  is [[dense set|dense]] in the complex plane. A result of the Great Picard Theorem is that any entire, non-polynomial function attains all possible complex values infinitely often, with at most one exception.

The "single exception" is needed in both theorems, as demonstrated here: 
*[[exponential function|e<sup>z</sup>]] is an entire non-constant function that is never 0,
*<math display="inline">e^{\frac{1}{z}}</math>  has an essential singularity at 0, but still never attains 0 as a value.

==Proof==

===Little Picard Theorem===
Suppose <math display="inline">f: \mathbb{C}\to\mathbb{C}</math>  is an entire function that omits two values <math display="inline">z_0</math> and <math display="inline">z_1
</math>. Then <math display="inline">\frac{f(z)-z_0}{z_1 - z_0}</math> is also entire and we may assume without loss of generality that <math display="inline">z_0 = 0</math> and <math display="inline">z_1=1</math>.

Because <math display="inline">\mathbb{C}</math> is [[Simply connected space|simply connected]] and the range of <math display="inline">f</math>  omits <math display="inline">0
</math> , ''f'' has a [[Complex_logarithm#Logarithms_of_holomorphic_functions|holomorphic logarithm]]. Let <math display="inline">g</math>  be an entire function such that <math display="inline">f(z)=e^{2\pi ig(z)}</math>. Then the range of <math display="inline">g</math> omits all integers. By a similar argument using the [[quadratic formula]], there is an entire function ''<math display="inline">h</math>'' such that <math display="inline">g(z)=\cos(h(z))</math>. Then the range of <math display="inline">h</math> omits all [[complex number]]s of the form <math display="inline">2\pi n \pm i \cosh^{-1}(m)</math>, where <math display="inline">n
</math> is an integer and <math display="inline">m</math> is a nonnegative integer.

By [[Bloch's_theorem_(complex_variables)#Landau's_theorem|Landau's theorem]], if <math display="inline">h'(w) \ne 0</math>, then for all <math display="inline">{R > 0}</math>, the range of <math display="inline">h</math> contains a disk of radius <math display="inline">|h'(w)| R/72</math>. But from above, any sufficiently large disk contains at least one number that the range of ''h'' omits. Therefore <math display="inline">h'(w)=0</math> for all <math display="inline">w</math>. By the [[fundamental theorem of calculus]], <math display="inline">h</math> is constant, so <math display="inline">f</math> is constant.

===Great Picard Theorem===
{{Collapse top|title=Proof of the Great Picard Theorem}}
Suppose ''f'' is an analytic function on the [[punctured disk]] of radius ''r'' around the point ''w'', and that ''f'' omits two values ''z''<sub>0</sub> and ''z''<sub>1</sub>. By considering (''f''(''p'' + ''rz'') − ''z''<sub>0</sub>)/(''z''<sub>1</sub> − ''z''<sub>0</sub>) we may assume without loss of generality that ''z''<sub>0</sub> = 0, ''z''<sub>1</sub> = 1, ''w'' = 0, and ''r'' = 1. 

The function ''F''(''z'') = ''f''(''e''<sup>−''z''</sup>) is analytic in the right half-plane Re(''z'') &gt; 0. Because the right half-plane is simply connected, similar to the proof of the Little Picard Theorem, there are analytic functions ''G'' and ''H'' defined on the right half-plane such that ''F''(''z'') = ''e''<sup>2π''iG''(''z'')</sup> and ''G''(''z'') = cos(''H''(''z'')). For any ''w'' in the right half-plane, the open disk with radius Re(''w'') around ''w'' is contained in the domain of ''H''. By Landau's theorem and the observation about the range of ''H'' in the proof of the Little Picard Theorem, there is a constant ''C'' &gt; 0 such that |''H''′(''w'')| ≤ ''C'' / Re(''w''). Thus, for all real numbers ''x'' ≥ 2 and 0 ≤ ''y'' ≤ 2π,
:::<math>|H(x+iy)|=\left|H(2+iy)+\int_2^xH'(t+iy)\,\mathrm{d}t\right|\le|H(2+iy)|+\int_2^x\frac{C}{t}\,\mathrm{d}t\le A\log x,</math>
where ''A'' &gt; 0 is a constant. So |''G''(''x'' + ''iy'')| ≤ ''x''<sup>''A''</sup>.

Next, we observe that ''F''(''z'' + 2π''i'') = ''F''(''z'') in the right half-plane, which implies that ''G''(''z'' + 2π''i'') − ''G''(''z'') is always an integer. Because ''G'' is continuous and its domain is [[Connected space|connected]], the difference ''G''(''z'' + 2π''i'') − ''G''(''z'') = ''k'' is a constant. In other words, the function ''G''(''z'') − ''kz'' / (2π''i'') has period 2π''i''. Thus, there is an analytic function ''g'' defined in the punctured disk with radius ''e''<sup>−2</sup> around 0 such that ''G''(''z'') − ''kz'' / (2π''i'') = ''g''(''e''<sup>−''z''</sup>).

Using the bound on ''G'' above, for all real numbers ''x'' ≥ 2 and 0 ≤ ''y'' ≤ 2π,
::<math>\left|G(x+iy)-\frac{k(x+iy)}{2\pi i}\right|\le x^A+\frac{|k|}{2\pi}(x+2\pi)\le C'x^{A'}</math>
holds, where ''A''′ &gt; ''A'' and ''C''′ &gt; 0 are constants. Because of the periodicity, this bound actually holds for all ''y''. Thus, we have a bound |''g''(''z'')| ≤ ''C''′(−log|''z''|)<sup>''A''′</sup> for 0 &lt; |''z''| &lt; ''e''<sup>−2</sup>. By [[Removable_singularity#Riemann's_theorem|Riemann's theorem on removable singularities]], ''g'' extends to an analytic function in the open disk of radius ''e''<sup>−2</sup> around 0.

Hence, ''G''(''z'') − ''kz'' / (2π''i'') is bounded on the half-plane Re(''z'') ≥ 3. So ''F''(''z'')''e''<sup>−''kz''</sup> is bounded on the half-plane Re(''z'') ≥ 3, and
''f''(''z'')''z''<sup>''k''</sup> is bounded in the punctured disk of radius ''e''<sup>−3</sup> around 0. By Riemann's theorem on removable singularities, ''f''(''z'')''z''<sup>''k''</sup> extends to an analytic function in the open disk of radius ''e''<sup>−3</sup> around 0. Therefore, ''f'' does not have an essential singularity at 0.

Therefore, if the function ''f'' has an essential singularity at 0, the range of ''f'' in any open disk around 0 omits at most one value. If ''f'' takes a value only finitely often, then in a sufficiently small open disk around 0, ''f'' omits that value. So ''f''(''z'') takes all possible complex values, except at most one, infinitely often.
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==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".

==Notes==
<references/>

==References==
* {{cite book|first=John B.|last=Conway|authorlink=John B. Conway|year=1978|title=Functions of One Complex Variable I|edition=2nd|publisher=Springer|isbn=0-387-90328-3}}
*{{cite web|url= http://people.reed.edu/~jerry/311/picard.pdf|last=Shurman|first=Jerry|title=Sketch of Picard's Theorem|accessdate=2010-05-18}}

[[Category:Theorems in complex analysis]]