Editing Analytic continuation (section)

==Initial discussion==
[[Image:Imaginary log analytic continuation.png|316px|right|thumb|Analytic continuation of natural logarithm (imaginary part)]]

Suppose ''f'' is an [[analytic function]] defined on a non-empty [[open set|open subset]] ''U'' of the [[complex plane]] {{nowrap|<math>\Complex</math>. }} If ''V'' is a larger open subset of {{nowrap|<math>\Complex</math>,}} containing ''U'', and ''F'' is an analytic function defined on ''V'' such that

:<math>F(z) = f(z) \qquad \forall z \in U, </math>

then ''F'' is called an analytic continuation of ''f''. In other words, the [[Restriction (mathematics)|restriction]] of ''F'' to ''U'' is the function ''f'' we started with.

Analytic continuations are unique in the following sense: if ''V'' is the [[connectedness|connected]] domain of two analytic functions ''F''<sub>1</sub> and ''F''<sub>2</sub> such that ''U'' is contained in ''V'' and for all ''z'' in ''U''

:<math>F_1(z) = F_2(z) = f(z),</math>

then

:<math>F_1 = F_2</math>

on all of ''V''. This is because ''F''<sub>1</sub>&nbsp;−&nbsp;''F''<sub>2</sub> is an analytic function which vanishes on the open, connected domain ''U'' of ''f'' and hence must vanish on its entire domain. This follows directly from the [[identity theorem]] for [[holomorphic function]]s.