Editing Picard theorem (section)

==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===
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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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