Editing Holomorphic function (section)

== Examples ==
All [[polynomial]] functions in {{tmath|z}}  with complex [[coefficient]]s are [[entire function]]s (holomorphic in the whole complex plane {{tmath|\C}}), and so are the [[exponential function#Complex plane|exponential function]] {{tmath|\exp z}} and the [[trigonometric functions]] {{tmath|1= \cos{z} = \tfrac{1}{2} \bigl( \exp(+iz) + \exp(-iz)\bigr)}} and {{tmath|1= \sin{z} = -\tfrac{1}{2} i \bigl(\exp(+iz) - \exp(-iz)\bigr)}} (cf. [[Euler's formula]]). The [[principal branch]] of the [[complex logarithm]] function {{tmath|\log z}} is holomorphic on the domain {{tmath|\C \smallsetminus \{ z \in \R : z \le 0\} }}. The [[square root#Principal square root of a complex number|square root]] function can be defined as {{tmath|\sqrt{z} \equiv \exp \bigl(\tfrac{1}{2} \log z\bigr) }} and is therefore holomorphic wherever the logarithm {{tmath|\log z}} is. The [[multiplicative inverse#Complex numbers|reciprocal function]] {{tmath|\tfrac{1}{z} }} is holomorphic on {{tmath| \C \smallsetminus \{ 0 \} }}. (The reciprocal function, and any other [[rational function]], is [[meromorphic function|meromorphic]] on {{tmath|\C}}.)

As a consequence of the [[Cauchy–Riemann equations]], any real-valued holomorphic function must be [[constant function|constant]]. Therefore, the [[absolute value#Complex numbers|absolute value]] {{nobr|<math>|z|</math>,}} the [[argument (complex analysis)|argument]] {{tmath|\arg z}}, the [[Complex number#Notation|real part]] {{tmath|\operatorname{Re}(z)}} and the [[Complex number#Notation|imaginary part]] {{tmath|\operatorname{Im}(z)}} are not holomorphic. Another typical example of a [[continuous function]] which is not holomorphic is the [[complex conjugate]] {{tmath|\bar z.}} (The complex conjugate is [[antiholomorphic function|antiholomorphic]].)