User contributions for Vipul

16 April 2008

• 22:3522:35, 16 April 2008 diff hist +513‎ N Fundamental theorem of algebra ‎ New page: ==Definition== Let <math>p:\mathbb{C} \to \mathbb{C}</math> be a polynomial function; in other words, there exist <math>a_0, a_1, \ldots, a_n \in \mathbb{C}</math> such that for all <...
• 22:3222:32, 16 April 2008 diff hist +437‎ N Cauchy integral formula for derivatives ‎ New page: ==Statement== Suppose <math>U</math> is a domain in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a holomorphic function. Suppose <math>\gamma</math> is the circl...
• 22:2622:26, 16 April 2008 diff hist +705‎ N Cauchy estimates for derivatives ‎ New page: ==Statement== Let <math>U</math> be a domain in <math>\mathbb{C}</math> and <math>f: U \to \mathbb{C}</math> be a holomorphic function. Suppose there exists a nonnegative real con...
• 21:5621:56, 16 April 2008 diff hist +29‎ N Goursat lemma ‎ Redirecting to Goursat's lemma
• 21:5421:54, 16 April 2008 diff hist +673‎ N Convex domain ‎ New page: ==Definition== ===In one dimension=== A domain (open connected subset) in <math>\mathbb{C}</math> is termed '''convex''' if given any two points in the domain, the entire line segmen...
• 21:5021:50, 16 April 2008 diff hist +506‎ N Logarithmically convex domain ‎ New page: ==Definition== A domain (connected open subset) in <math>\mathbb{C}</math> is termed '''logarithmically convex''' or '''log-convex''' if we can choose a branch of the logarithm th...
• 21:4721:47, 16 April 2008 diff hist +43‎ N Log-convex domain ‎ Redirecting to Logarithmically convex domain
• 21:4521:45, 16 April 2008 diff hist +2,038‎ N Nowhere zero holomorphic function on simply connected domain admits holomorphic logarithm ‎ New page: ==Definition== Suppose <math>U</math> is a simply connected domain in <math>\mathbb{C}</math>: in other words <math>U</math> is an open subset of <math>\mathbb{C}</math> such that eve...
• 21:2521:25, 16 April 2008 diff hist +784‎ N Conformal mapping ‎ New page: ==Definition== Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math>. A map <math>f: U \to \mathbb{C}</math> (or to a subset of <math>\mathbb{C}</math> is termed a '''confor...
• 21:1921:19, 16 April 2008 diff hist +85‎ Function holomorphic at a point ‎No edit summary
• 21:1921:19, 16 April 2008 diff hist 0‎ m Function continuously real-differentiable at a point ‎ Function continuously differentiable at a point moved to Function continuously real-differentiable at a point
• 21:1921:19, 16 April 2008 diff hist +66‎ N Function continuously differentiable at a point ‎ Function continuously differentiable at a point moved to Function continuously real-differentiable at a point
• 21:1921:19, 16 April 2008 diff hist +852‎ N Function continuously real-differentiable at a point ‎ New page: {at-point function property|continuously real-differentiable function}} ==Definition== ===In one dimension over complex numbers=== Let <math>U \subset \mathbb{C}</math> be an open subse...
• 21:1321:13, 16 April 2008 diff hist +667‎ N Function holomorphic at a point ‎ New page: {{at-point function property|holomorphic function}} ==Definition== ===In one dimension=== Let <math>U</math> be an open subset (without loss of generality, an open connected subset, i.e...
• 21:1121:11, 16 April 2008 diff hist +8‎ Function complex-analytic at a point ‎ →‎In one dimension
• 21:1021:10, 16 April 2008 diff hist +873‎ N Function complex-analytic at a point ‎ New page: {{at-point function property|holomorphic function}} ==Definition== ===In one dimension=== Suppose <math>U</math> is an open subset (without loss of generality, a domain, i.e. an open...
• 21:0621:06, 16 April 2008 diff hist −5‎ Template:At-point function property ‎No edit summary
• 21:0621:06, 16 April 2008 diff hist +21‎ Function complex-differentiable at a point ‎No edit summary
• 21:0521:05, 16 April 2008 diff hist +354‎ N Template:At-point function property ‎ New page: {{quotation|''This article defines a property that can be evaluated for a function on a (particular kind of) set, and a point in that set. If the function satisfies this property at every ...
• 21:0421:04, 16 April 2008 diff hist +31‎ Function complex-differentiable at a point ‎No edit summary
• 20:5820:58, 16 April 2008 diff hist +375‎ N Function complex-differentiable at a point ‎ New page: ==Definition== ===In one dimension=== Let <math>U</math> be a domain in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> be a function. Let <math>z_0 \in U</math>. We say ...
• 20:5620:56, 16 April 2008 diff hist +449‎ Biholomorphically equivalent domains ‎No edit summary
• 18:2618:26, 16 April 2008 diff hist +8‎ Biholomorphically equivalent domains ‎No edit summary
• 18:2518:25, 16 April 2008 diff hist +593‎ N Biholomorphically equivalent domains ‎ New page: ==Definition== Let <math>U,V \subset \mathbb{C}^n</math> be domains (open connected subset). We say that <math>U,V</math> are '''biholomorphically equivalent''' if there exists a [[h...
• 18:1918:19, 16 April 2008 diff hist +1,052‎ Holomorphic function ‎No edit summary
• 17:5817:58, 16 April 2008 diff hist +188‎ Complex differential of a complex-valued function ‎No edit summary
• 17:5517:55, 16 April 2008 diff hist +56‎ N Complex-differentiable function ‎ Redirecting to Function complex-differentiable at a point

14 April 2008

• 22:5622:56, 14 April 2008 diff hist +962‎ Complex-analytic implies holomorphic ‎No edit summary
• 22:2922:29, 14 April 2008 diff hist +31‎ Complex-analytic implies holomorphic ‎No edit summary
• 20:2420:24, 14 April 2008 diff hist +654‎ N Complex-analytic implies holomorphic ‎ New page: ==Statement== Suppose <math>U \subset \mathbb{C}</math> is a domain and <math>f:U \to \mathbb{C}</math> is a complex-analytic function: for every point <math>z_0 \in U</math>, there e...
• 20:2020:20, 14 April 2008 diff hist +22‎ N Template:Fillin ‎ New page: ''Fill this in later''
• 20:1920:19, 14 April 2008 diff hist +551‎ N Template:Further ‎ New page: :''Further information: {{{1}}}''<noinclude> == Usage == * Different from Template:See (which it supersedes), this template takes a single parameter of any length. * Use commas and j...
• 01:0901:09, 14 April 2008 diff hist +694‎ N Looman-Menchoff theorem ‎ New page: ==Statement== Let <math>U</math> be a domain in the complex numbers, and <math>f:U \to \mathbb{C}</math> be a continuous function, such that: * For any point <math>z \in U</math>, th...
• 00:5200:52, 14 April 2008 diff hist +1,244‎ N Holomorphic implies complex-analytic ‎ New page: {{definition equivalence|holomorphic function}} ==Statement== Suppose <math>U</math> is a domain in <math>\mathbb{C}</math>, and <math>f: U \to \mathbb{C}</math> is holomorphic: it i...
• 00:4500:45, 14 April 2008 diff hist +1,473‎ N Cauchy integral formula ‎ New page: ==Statement== ===For a circle with respect to its center=== Suppose <math>U</math> is a domain in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic funct...
• 00:3400:34, 14 April 2008 diff hist +1,123‎ N Goursat's integral lemma ‎ New page: ==Statement== ===For a triangle=== Suppose <math>U</math> is a domain in <math>\mathbb{C}</math>, and <math>f:U \to \mathbb{C}</math> is a holomorphic function. Suppose <math>\tr...
• 00:2500:25, 14 April 2008 diff hist +447‎ N Fundamental theorem of complex calculus ‎ New page: ==Statement== Suppose <math>f</math> is a holomorphic function on a domain <math>U \subset \mathbb{C}</math>, and <math>g := f'</math> is its complex differential. Suppose <math>\...

13 April 2008

• 23:3823:38, 13 April 2008 diff hist +409‎ N Domain ‎ New page: ==Definition== A '''domain in the complex numbers''', or simply a '''domain''', is a connected open subset of the complex numbers <math>\mathbb{C}</math>. Complex analysis is the study of...
• 23:3623:36, 13 April 2008 diff hist +947‎ N Differential 1-form associated with a complex-valued function ‎ New page: ==Definition== Let <math>U</math> be an open subset of <math>\mathbb{C}</math>, and <math>f:U \to \mathbb{C}</math> be a continuous function. The differential 1-form associated with <math...
• 22:0922:09, 13 April 2008 diff hist +180‎ Integral of a complex-valued function along a curve ‎No edit summary
• 22:0122:01, 13 April 2008 diff hist +373‎ N Integral of a complex-valued function along a curve ‎ New page: ==Definition== Suppose <math>\gamma:[a,b] \to \mathbb{C}</math> is a smooth curve whose image lies inside an open subset <math>U \subset \mathbb{C}</math>. Suppose <math>f:U \to \mathbb{C...
• 21:1621:16, 13 April 2008 diff hist +149‎ N Template:Quotation ‎ New page: <blockquote class="toccolours" style="float:none; padding: 10px 15px 10px 15px; display:table;"> {{{1<noinclude>|  {{Lorem}}</noinclude>}}}
• 21:1621:16, 13 April 2008 diff hist +35‎ Cauchy-Riemann differential equations ‎ →‎Definition
• 21:0521:05, 13 April 2008 diff hist +1,052‎ N Cauchy-Riemann differential equations ‎ New page: ==Definition== Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a function. Write <math>f</math> in terms of its real and imagina...
• 20:5020:50, 13 April 2008 diff hist +51‎ N Cauchy-Riemann equations ‎ Redirecting to Cauchy-Riemann differential equations
• 20:4920:49, 13 April 2008 diff hist +1,814‎ N Complex differential of a complex-valued function ‎ New page: ==Definition== ===Definition as a general limit=== Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a function. Let <math>z_0 \i...
• 20:0120:01, 13 April 2008 diff hist +430‎ N Entire function ‎ New page: ==Definition== An '''entire function''' is defined in the following equivalent ways: * It is a holomorphic function on the set of all complex numbers * It is a global analytic functi...
• 19:4219:42, 13 April 2008 diff hist +554‎ N Radius of convergence ‎ New page: ==Definition== Consider the power series about a point <math>z_0 \in \mathbb{C}</math> with coefficients <math>a_n \in \mathbb{C}</math>: <math>\sum a_n(z - z_0)^n</math> The '''radius ...
• 19:2919:29, 13 April 2008 diff hist +1‎ Holomorphic function ‎ →‎Definition with symbols
• 19:2819:28, 13 April 2008 diff hist +1,091‎ N Holomorphic function ‎ New page: ==Definition== ===Definition with symbols=== Let <math>\Omega</math> be an open subset of <math>\mathbb{C}</math>. A function <math>f:U \to \mathbb{C}</math> is termed a '''holomorphic f...