Holomorphic function
Examples

All polynomial functions in z {\displaystyle z} with complex coefficients are entire functions (holomorphic in the whole complex plane C {\displaystyle \mathbb {C} }), and so are the exponential function exp z {\displaystyle \exp z} and the trigonometric functions cos z = 1 2 (exp (+ i z) + exp (− i z)) {\displaystyle \cos {z}={\tfrac {1}{2}}{\bigl (}\exp(+iz)+\exp(-iz){\bigr)}} and sin z = − 1 2 i (exp (+ i z) − exp (− i z)) {\displaystyle \sin {z}=-{\tfrac {1}{2}}i{\bigl (}\exp(+iz)-\exp(-iz){\bigr)}} (cf. Euler's formula). The principal branch of the complex logarithm function log z {\displaystyle \log z} is holomorphic on the domain C ∖ { z ∈ R: z ≤ 0 } {\displaystyle \mathbb {C} \smallsetminus \{z\in \mathbb {R}:z\leq 0\}}. The square root function can be defined as z ≡ exp (1 2 log z) {\displaystyle {\sqrt {z}}\equiv \exp {\bigl (}{\tfrac {1}{2}}\log z{\bigr)}} and is therefore holomorphic wherever the logarithm log z {\displaystyle \log z} is. The reciprocal function 1 z {\displaystyle {\tfrac {1}{z}}} is holomorphic on C ∖ { 0 } {\displaystyle \mathbb {C} \smallsetminus \{0\}}. (The reciprocal function, and any other rational function, is meromorphic on C {\displaystyle \mathbb {C} }.)

As a consequence of the Cauchy–Riemann equations, any real-valued holomorphic function must be constant. Therefore, the absolute value | z | {\displaystyle |z|}, the argument arg z {\displaystyle \arg z}, the real part Re (z) {\displaystyle \operatorname {Re} (z)} and the imaginary part Im (z) {\displaystyle \operatorname {Im} (z)} are not holomorphic. Another typical example of a continuous function which is not holomorphic is the complex conjugate z ¯. {\displaystyle {\bar {z}}.} (The complex conjugate is antiholomorphic.)
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