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If the following limit is a finite complex number, then that complex number equals the residue at z_0: \lim_{z \to z_0} (z - z_0)f(z) ... ...

* It is a holomorphic function on the set of all complex numbers globally convergent power series about any complex number ... ...

===Definition in terms of complex integrals=== the loops comprising c. Then, the winding number of c about z_0, denoted n ... ...

===For an open subset in the complex numbers=== e. maps all elements of U to the same complex number) or an open map: the ... ...

is a domain and f:U \to \mathbb{C} is a complex-analytic function: for every ... Then, f is a holomorphic function: it is complex-differentiable, and ... ...

It is defined as follows. If z = x + iy is a complex number with x,y being respectively the real part and imaginary part, then: |z| = \sqrt{x^2 + y^2} ... ...

Suppose (z_n) is a sequence of (possibly repeating) complex numbers that does not cluster in \mathbb{C}: in other words, |z_n| \to \infty if ... ...

The property of being complex-differentiable at a point is equivalent ... Moreover the complex differential of f at z_0 is the same complex ... ...

Suppose c \in \mathbb{C} is a complex number, and U is a domain in \mathbb{C}. Then, if D is a disk centered at z_0, and z is any point in the ... ...

* If g is a primitive of f'/f, then we can find a complex number c such that g + c is a holomorphic logarithm of f ==Related facts== ... ...

* Winding number version of Cauchy integral formula * Homotopy-invariance ... that allows the exchange of a real and a complex integral when the total ... ...

lemma, f is a rotation i.e. multiplication by a complex number of unit modulus. Rotations are fractional linear transformations, so we're done. ... ...