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  • Ernst Lindel\"of | planetmath.org
    jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A13A18 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Ernst Leonard Lindelöf 1870 3 7 à 1946 6 3 was born in Helsinki Finland Professor of mathematics in University of Helsinki 1903 1938 Lindelöf was the founder of the internationally noted Finnish school of function theory Rolf Nevanlinna and Lars Ahlfors were his pupils Most studies of Lindelöf were on the complex analysis theory of differential equations e g Picard Lindelöf theorem and topology see e g Lindelöf space One of them Mémoire sur la théorie des fonctions entières de genre fini 1902 complements substantially the theory of entire functions also Mémoire sur certaines inégalités dans la théorie des fonctions monogènes 1908 containing a new simple proof of Picard s theorem was important Lindelöf has made excellent textbooks of mathematics Le calcul des résidus et ses applications à la théorie des fonctions 1905 Johdatus korkeampaan analyysiin Introduction to the higher analysis 1912

    Original URL path: http://www.planetmath.org/ernstlindelof (2016-04-25)
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  • errors can cancel each other out | planetmath.org
    3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit errors can cancel each other out If one uses the change of variable tan x t d x d t 1 t 2 cos 2 x 1 1 t 2 formulae sequence assign x t formulae sequence d x d t 1 superscript t 2 superscript 2 x 1 1 superscript t 2 displaystyle tan x t quad dx frac dt 1 t 2 quad cos 2 x frac 1 1 t 2 1 for finding the value of the definite integral I π 4 3 π 4 d x 2 cos 2 x 1 assign I superscript subscript π 4 3 π 4 d x 2 superscript 2 x 1 I int frac pi 4 frac 3 pi 4 frac dx 2 cos 2 x 1 the following calculation looks appropriate and faultless I 1 1 d t 3 t 2 1 3 1 1 arctan t 3 1 3 5 π 6 π 6 2 π 3 3 I superscript subscript 1 1 d t 3 superscript t 2 1 3 superscript subscript normal 1 1 t 3 1 3 5 π 6 π 6 2 π 3 3 displaystyle I int 1 1 frac dt 3 t 2 frac 1 sqrt 3 operatornamewithlimits Big 1 quad 1 arctan frac t sqrt 3 frac 1 sqrt 3 left frac 5 pi 6 frac pi 6 right frac 2 pi 3 sqrt 3 2 The result is quite right Unfortunately the calculation contains two errors the effects of which cancel each other out The crucial error in 2 is using the substitution 1 when tan x x tan x is discontinuous in the point x π 2 x π 2 x frac pi 2 on the interval π 4 3 π 4 π 4 3 π 4 frac pi 4 frac 3 pi 4 of integration The error is however canceled out by the second error using the value 5 π 6 5 π 6 frac 5 pi 6 for arctan 1 3 1 3 arctan frac 1 sqrt 3 when the right value were π 6 π 6 frac pi 6 the values of arctan lie only between π 2 π 2 frac pi 2 and π 2 π 2 frac pi 2 see cyclometric functions The value 5 π 6 5 π 6 frac 5 pi 6 belongs to a different branch of the inverse tangent function than π 6 π 6 frac pi 6 parts of two distinct branches cannot together form the antiderivative which must be continuous What were a right way to calculate I I I The universal trigonometric substitution produces an awkward integrand 2 2 t 2 3 2 t 2 3 t 4 2 2 superscript t 2 3 2 superscript

    Original URL path: http://www.planetmath.org/errorscancanceleachotherout (2016-04-25)
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  • estimating theorem of contour integral | planetmath.org
    2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A06 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A06 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A97D70 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A97D70 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql

    Original URL path: http://www.planetmath.org/estimatingtheoremofcontourintegral (2016-04-25)
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  • estimation of index of intersection subgroup | planetmath.org
    3E 7B msc 3A30A99 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A30A99 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit estimation of index of intersection subgroup Theorem If H 1 H 2 H n subscript H 1 subscript H 2 normal subscript H n H 1 H 2 ldots H n are subgroups of G G G then G i 1 n H i i 1 n G H i fragments fragments normal G normal superscript subscript i 1 n subscript H i normal superscript subscript product i 1 n fragments normal G normal subscript H i normal normal left G bigcap i 1 n H i right leqq prod i 1 n G H i Proof We prove here only the case n 2 n 2 n 2 the general case may be handled by the induction Let H 1 H 2 K assign subscript H 1 subscript H 2 K H 1 cap H 2 K Let R R R be the set of the right cosets of K K K and R i subscript R i R i the set of the right cosets of H i subscript H i H i i 1 2 i 1 2 i 1 2 Define the relation ϱ ϱ varrho from R R R to R 1 R 2 subscript R 1 subscript R 2 R 1 times R 2 as ϱ K x H 1 x H 2 x x G fragments ϱ assign fragments normal fragments normal K x normal fragments normal subscript H 1 x normal subscript H 2 x normal normal normal x G normal normal varrho left Kx H 1 x H 2 x right vdots x in G We then have the equivalent conditions K x K y K x K y Kx Ky x y 1 K x superscript y 1 K xy 1 in K x y 1 H 1 x y 1 H 2 fragments x superscript y 1 subscript

    Original URL path: http://www.planetmath.org/estimationofindexofintersectionsubgroup (2016-04-25)
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  • Euclid's coefficients | planetmath.org
    2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A20D99 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Euclid s coefficients The following program is based on Euclid s algorithm and it determines Euclid s coefficients t t t and d d d of the integers x x x and y y y INPUT x x x y y y positive integers m x assign m x m x n y assign n y n y b 0 assign b 0 b 0 d 1 assign d 1 d 1 p 1 assign p 1 p 1 t 0 assign t 0 t 0 WHILE m 0 m 0 m neq 0 DO BEGIN q n assign q n q n DIV m m m integer division h m assign h m h m m n q m assign m n q m m n qm n h assign n h n h h b assign h b h b b d q b assign b d q b b d qb d h assign d h d h h p assign h p h p p t q p assign p t q p p t qp t h assign t h t h END WRITE The gcd of the numbers x x x and y y y is n t x d

    Original URL path: http://www.planetmath.org/euclidscoefficients (2016-04-25)
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  • Euclidean axiom by Hilbert | planetmath.org
    2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A03 04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A03 04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Euclidean axiom by Hilbert In Hilbert s Grundlagen der Geometrie Foundations of Geometry the original edition in 1899 there is the following argumentation Let α α alpha be an arbitrary plane a a a a line in α α alpha and A A A a point in α α alpha which lies outside a a a If we draw in α α alpha a line c c c which passes through A A A and intersects a a a and then through A A A a line b b b such that the line c c c intersects the lines a a a b b b with equal alternate interior angles unter gleichen Gegenwinkeln then it follows easily from the theorem on the outer angles that the lines a a a b b b have no common point i e in a plane α α alpha one can always draw otside a line a a a another line which does not intersect the line a a a The Parallel Axiom reads now IV Euclidean Axiom Let a a a be an arbitrary line and A A A be a point outside a a a then in the plane determined by a a a and A A A there exists at most one line which passes through A A A and does not intersect a a a Explanation According the the preceding text and on grounds of the Parallel Axiom we realize that there is one and only

    Original URL path: http://www.planetmath.org/euclideanaxiombyhilbert (2016-04-25)
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  • Euclidean geometry of space | planetmath.org
    sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51 01 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51 01 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Euclidean geometry of space Please help to list the existing and future entries on spatial geometry Basic objects 1 point 2 line ray line segment

    Original URL path: http://www.planetmath.org/euclideangeometryofspace (2016-04-25)
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  • Euler numbers | planetmath.org
    in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Euler numbers E n subscript E n E n have the generating function 1 cosh x 1 x displaystyle frac 1 cosh x such that 1 cosh x n 0 E n n x n fragments 1 x normal superscript subscript n 0 subscript E n n superscript x n normal frac 1 cosh x sum n 0 infty frac E n n x n They are integers but have no simple expression for calculating them Their only regularities are that the numbers with odd index are all 0 and that sgn E 2 m 1 m for m 0 1 2 formulae sequence sgn subscript E 2 m superscript 1 m for m 0 1 2 normal mbox sgn E 2m 1 m qquad mbox for quad m 0 1 2 ldots The Euler number have intimate relation to the Bernoulli numbers The first Euler numbers with even index are E 0 1 E 2 1 E 4 5 E 6 61 E 8 1385 E 10 50521 formulae sequence subscript E 0 1 formulae sequence subscript E 2 1 formulae sequence subscript E 4 5 formulae sequence subscript E 6 61 formulae sequence subscript E 8 1385 subscript E 10 50521 E 0 1 quad E 2 1 quad E 4 5 quad E 6 61 quad E 8 1385 quad E 10 50521 One can by hand determine Euler numbers by performing the division of 1 by the Taylor series of hyperbolic cosine cf Taylor series via division and Taylor series of hyperbolic functions Since cosh i x cos x i x x cosh ix cos x the division 1 cos x normal

    Original URL path: http://www.planetmath.org/eulernumbers (2016-04-25)
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