Revision #1065 → #2931 · back to history
modifiedCauchy's integral formula507734b20770
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| anchors | [{"section":"Theorem","snippet":"suppose the closed disk"},{"type":"math_alttext","value":"{\\displaystyle D={\\bigl \\{}z\\in \\mathbb {C} :|z-z_{0}|\\leq r{\\bigr \\}}}"},{"type":"math_alttext","value":"{\\displaystyle f(a)={\\frac {1}{2\\pi i}}\\oint _{\\gamma }{\\frac {f(z)}{z-a}}\\,dz.}"}] | — |
| note | Gives ∮ (z-w)⁻¹•f z = 2πi•f w under continuity on the closed disc and differentiability on the interior, which is more general than the article's holomorphic-on-an-open-set hypothesis. | Gives ∮ (z-w)⁻¹•f z = 2πi•f w under continuity on the closed disc and differentiability on the interior, more general than the article's holomorphic-on-open-set hypothesis. |
modifiedAnalyticity of holomorphic functions810900af116e
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| anchors | [{"section":"Theorem","snippet":"it follows that holomorphic functions are analytic"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {1}{z-a}}={\\frac {1+{\\frac {a}{z}}+\\left({\\frac {a}{z}}\\right)^{2}+\\cdots }{z}},}"},{"type":"math_alttext","value":"{\\displaystyle f^{(n)}(a)={\\frac {n!}{2\\pi i}}\\oint _{\\gamma }{\\frac {f(z)}{\\left(z-a\\right)^{n+1}}}\\,dz.}"}] | — |
| label | Cauchy's differentiation formula (analyticity) | Analyticity of holomorphic functions |
| note | States that a function differentiable on an open set in ℂ is analytic (admits a power series) on that set. | A function differentiable on an open set in ℂ is analytic (admits a convergent power series expansion) on that set. |
| provenance | ai | ai-moderated |
addedCauchy's differentiation formula (higher derivatives)e09a866bba07
modifiedNot every boundary function arisesd7c3253e75a7
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| note | This remark about boundary functions not all extending holomorphically has no corresponding statement in Mathlib. | This informal remark about boundary functions not all extending holomorphically has no corresponding statement in Mathlib. |
modifiedEvaluating a contour integralb19677ef9069
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| anchors | [{"section":"Example","snippet":"the circle of radius 2"},{"type":"math_alttext","value":"{\\displaystyle g(z)={\\frac {z^{2}}{z^{2}+2z+2}},}"}] | — |
modifiedHolomorphic implies infinitely differentiable8a254d2e5e5d
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| anchors | [{"section":"Consequences","snippet":"a function that is holomorphic in an open set is in fact infinitely differentiable there"},{"type":"math_alttext","value":"{\\displaystyle f(\\zeta )={\\frac {1}{2\\pi i}}\\int _{C}{\\frac {f(z)}{z-\\zeta }}\\,dz.}"}] | — |
| note | Shows a function differentiable on an open set in ℂ is ContDiffOn of any order n (including ∞), i.e. infinitely differentiable there. | A function differentiable on an open set in ℂ is ContDiffOn of any order n (including ∞), i.e. infinitely differentiable there. |
modifiedUniform limit of holomorphic functionse94447040f05
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| note | Establishes that a locally uniform limit of holomorphic functions on an open set is holomorphic, generalizing the uniform-limit statement. | A locally uniform limit of holomorphic functions on an open set is holomorphic, generalizing the uniform-limit statement. |
addedMorera's theorem8311531ea9ee
modifiedCauchy's estimate506af5d4459e
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| anchors | [{"section":"Consequences","snippet":"satisfy Cauchy's estimate"},{"type":"math_alttext","value":"{\\displaystyle |a_{n}|\\leq r^{-n}\\sup _{|z|=r}|f(z)|.}"}] | — |
modifiedLiouville's theoremd83c2c179a87
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| note | Proves a complex-differentiable function with bounded range is constant, stated for Banach-space-valued maps and hence more general than entire ℂ→ℂ functions. | A complex-differentiable function with bounded range is constant, stated for Banach-space-valued maps and hence more general than entire ℂ→ℂ. |
modifiedGauss's mean-value theoremac7c19d2a6a8
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| anchors | [{"section":"Consequences","snippet":"derive Gauss's mean-value theorem"},{"type":"math_alttext","value":"{\\displaystyle f(z)={\\frac {1}{2\\pi }}\\int _{0}^{2\\pi }f(z+re^{i\\theta })\\,d\\theta .}"}] | — |
| note | States that the circle average of a holomorphic function equals its value at the center, i.e. Gauss's mean-value property. | The circle average of a holomorphic function equals its value at the center, i.e. Gauss's mean-value property. |
modifiedCauchy–Pompeiu formula5f119651b900
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| anchors | [{"section":"Smooth functions","snippet":"is a complex-valued C 1 function on the closure of"},{"type":"math_alttext","value":"{\\displaystyle f(\\zeta )={\\frac {1}{2\\pi i}}\\int _{\\partial D}{\\frac {f(z)\\,dz}{z-\\zeta }}-{\\frac {1}{\\pi }}\\iint _{D}{\\frac {\\partial f}{\\partial {\\bar {z}}}}(z){\\frac {dx\\wedge dy}{z-\\zeta }}.}"}] | — |
modifiedSolution of inhomogeneous Cauchy–Riemann equationsb998b3fd832c
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| anchors | [{"section":"Smooth functions","snippet":"to solve the inhomogeneous Cauchy–Riemann equations in"},{"type":"math_alttext","value":"{\\displaystyle d\\mu ={\\frac {1}{2\\pi i}}\\varphi \\,dz\\wedge d{\\bar {z}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\partial f}{\\partial {\\bar {z}}}}=\\varphi (z,{\\bar {z}}).}"}] | — |
modifiedCauchy kernel is a fundamental solution7a6cad57a75b
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| anchors | [{"section":"Smooth functions","snippet":"the Cauchy kernel is a fundamental solution of the Cauchy–Riemann equations"},{"type":"math_alttext","value":"{\\displaystyle k(z)=\\operatorname {p.v.} {\\frac {1}{z}}}"},{"type":"math_alttext","value":"{\\displaystyle f(\\zeta )={\\frac {1}{2\\pi i}}\\iint {\\frac {\\partial f}{\\partial {\\bar {z}}}}{\\frac {dz\\wedge d{\\bar {z}}}{z-\\zeta }},}"}] | — |
modifiedDistributional derivative of characteristic functionbaeba3e8ce74
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| anchors | [{"section":"Smooth functions","snippet":"the formula for the distributional derivative of the characteristic function"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\partial \\chi _{X}}{\\partial {\\bar {z}}}}={\\frac {i}{2}}\\oint _{\\partial X}\\,dz,}"}] | — |
modifiedCauchy formula on polydiscs8ea27145fd3d
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| anchors | [{"section":"Several variables","snippet":"the Cauchy integral formula can be generalized to polydiscs"},{"type":"math_alttext","value":"{\\displaystyle D=\\prod _{j=1}^{n}D_{j}.}"}] | — |
modifiedGreen's function of the derivative operator2e3aabb26cd8
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| anchors | [{"section":"In real algebras","snippet":"This particular derivative operator has a Green's function"},{"type":"math_alttext","value":"{\\displaystyle G\\left(\\mathbf {r} ,\\mathbf {r} '\\right)={\\frac {1}{S_{n}}}{\\frac {\\mathbf {r} -\\mathbf {r} '}{\\left|\\mathbf {r} -\\mathbf {r} '\\right|^{n}}}}"},{"type":"math_alttext","value":"{\\displaystyle \\nabla G\\left(\\mathbf {r} ,\\mathbf {r} '\\right)=\\delta \\left(\\mathbf {r} -\\mathbf {r} '\\right).}"}] | — |
modifiedMonogenic functionf994357f2ad5
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| anchors | [{"section":"In real algebras","snippet":"is called a monogenic function"},{"type":"math_alttext","value":"{\\displaystyle \\oint _{\\partial V'}G\\left(\\mathbf {r} ,\\mathbf {r} '\\right)\\;d\\mathbf {S} '\\;f\\left(\\mathbf {r} '\\right)=\\int _{V}\\left[\\nabla 'G\\left(\\mathbf {r} ,\\mathbf {r} '\\right)\\right]f\\left(\\mathbf {r} '\\right)=-\\int _{V}\\delta \\left(\\mathbf {r} -\\mathbf {r} '\\right)f\\left(\\mathbf {r} '\\right)\\;d\\mathbf {V} =-i_{n}f(\\mathbf {r} )}"},{"type":"math_alttext","value":"{\\displaystyle f(\\mathbf {r} )=-{\\frac {1}{i_{n}}}\\oint _{\\partial V}G\\left(\\mathbf {r} ,\\mathbf {r} '\\right)\\;d\\mathbf {S} \\;f\\left(\\mathbf {r} '\\right)=-{\\frac {1}{i_{n}}}\\oint _{\\partial V}{\\frac {\\mathbf {r} -\\mathbf {r} '}{S_{n}\\left|\\mathbf {r} -\\mathbf {r} '\\right|^{n}}}\\;d\\mathbf {S} \\;f\\left(\\mathbf {r} '\\right)}"}] | — |