**Companal, The Complex Analysis Wiki (pre-pre-alpha)**

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- * Winding number version of Cauchy integral formula * Homotopy-invariance
**...**that allows the exchange of a real and a complex integral when the total**...**3 KB (602 words) - 18:43, 12 September 2008 - 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**...**760 B (123 words) - 16:08, 12 September 2008 - The property of being complex-differentiable at a point is equivalent
**...**Moreover the complex differential of f at z_0 is the same complex**...**1 KB (169 words) - 19:10, 18 May 2008 - lemma, f is a rotation i.e. multiplication by a complex number of unit modulus. Rotations are fractional linear transformations, so we're done.
**...**2 KB (408 words) - 19:29, 12 September 2008 - * It is a holomorphic function on the set of all complex numbers globally convergent power series about any complex number
**...**430 B (68 words) - 19:12, 18 May 2008 - 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}
**...**590 B (102 words) - 19:16, 18 May 2008 - * 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==
**...**835 B (138 words) - 19:17, 18 May 2008 - ===For an open subset in the complex numbers=== e. maps all elements of U to the same complex number) or an open map: the
**...**597 B (109 words) - 19:17, 18 May 2008 - 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)
**...**1 KB (214 words) - 19:18, 18 May 2008 - 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
**...**664 B (113 words) - 19:19, 18 May 2008 - ===Definition in terms of complex integrals=== the loops comprising c. Then, the winding number of c about z_0, denoted n
**...**824 B (152 words) - 19:19, 18 May 2008 - 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
**...**1 KB (200 words) - 19:03, 12 September 2008