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  • all algebraic numbers in a sequence | planetmath.org
    module Primary tabs View active tab Coauthors PDF Source Edit all algebraic numbers in a sequence The beginning of the sequence of all algebraic numbers ordered as explained in the parent entry is as follows 0 1 1 2 1 2 i i 1 2 2 3 1 5 2 2 1 2 1 5 2 1 i 3 2 1 i 3 2 1 3 0 1 1 2 1 2 i i 1 2 2 3 1 5 2 2 1 2 1 5 2 1 i 3 2 1 i 3 2 1 3 0 1 1 2 frac 1 2 i i frac 1 2 2 3 frac 1 sqrt 5 2 sqrt 2 frac 1 sqrt 2 frac 1 sqrt 5 2 frac 1 i sqrt 3 2 frac 1 i sqrt 3 2 frac 1 3 i 2 i 2 i 2 i 2 1 3 1 i 3 2 1 i 3 2 1 5 2 1 2 2 1 5 2 3 i 2 i 2 i 2 i 2 1 3 1 i 3 2 1 i 3 2 1 5 2 1 2 2 1 5 2 3 normal i sqrt 2 frac i sqrt 2 frac i sqrt 2 i sqrt 2 frac 1 3 frac 1 i sqrt 3 2 frac 1 i sqrt 3 2 frac 1 sqrt 5 2 frac 1 sqrt 2 sqrt 2 frac 1 sqrt 5 2 3 ldots The first number corresponds to the algebraic equation x 0 x 0 x 0 the two following numbers to the equations x 1 0 plus or minus x 1 0 x pm 1 0 the six following to the equations x 2 0 plus or minus x 2 0 x pm 2 0 2

    Original URL path: http://www.planetmath.org/allalgebraicnumbersinasequence (2016-04-25)
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  • alternative definition of group | planetmath.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 3A03E10 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 3A03E10 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 alternative definition of group The below theorem gives three conditions that form alternative group postulates It is not hard to show that they hold in the group defined ordinarily Theorem Let the non empty set G G G satisfy the following three conditions I For every two elements a a a b b b of G G G there is a unique element a b a b ab of G G G II For every three elements a a a b b b c c c of G G G the equation a b c a b c a b c a b c ab c a bc holds III For every two elements a a a and b b b of G G G there exists at least one such element x x x and at least one such element y y y of G G G that x a a y b x a a y b xa ay b Then the set G G G forms a group Proof If a a a and b b b are arbitrary elements then there are at least one such e a subscript e a e a and such e b subscript e b e b that e a a a subscript e a a a e a a a and b e b b b subscript e b b be b b There are also such x x x and y y y that x b e a x b subscript e a xb e a and a y e b a y subscript e b ay e b Thus we have e a x b x b

    Original URL path: http://www.planetmath.org/alternativedefinitionofgroup (2016-04-25)
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  • analytic continuation of gamma function | planetmath.org
    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 3A20 00 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 3A08A99 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 3A08A99 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 analytic continuation of gamma function The last formula of the parent entry may be expressed as Γ z Γ z n z z 1 z 2 z n 1 normal Γ z normal Γ z n z z 1 z 2 normal z n 1 displaystyle Gamma z frac Gamma z n z z 1 z 2 cdots z n 1 1 According to the standard definition Γ z 0 e t t z 1 d t assign normal Γ z superscript subscript 0 superscript e t superscript t z 1 d t Gamma z int 0 infty e t t z 1 dt the left hand side of 1 is defined only in the right half plane ℜ z 0 z 0 Re z 0 whereas the expression Γ z n normal

    Original URL path: http://www.planetmath.org/analyticcontinuationofgammafunction (2016-04-25)
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  • analytic continuation of Riemann zeta to critical strip | planetmath.org
    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 3A33B15 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 3A33B15 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 The terms 1 n s e s log n 1 superscript 𝑛 𝑠 superscript 𝑒 𝑠 𝑛 see the general power of the series n 1 1 n s 1 1 2 s 1 3 s 1 4 s superscript subscript 𝑛 1 1 superscript 𝑛 𝑠 1 1 superscript 2 𝑠 1 superscript 3 𝑠 1 superscript 4 𝑠 1 defining the Riemann zeta function ζ s 𝜁 𝑠 for ℜ s 1 𝑠 1 are holomorphic in the whole s 𝑠 plane and the series converges uniformly in any closed disc of the half plane ℜ s 1 𝑠 1 let s σ i t 𝑠 𝜎 𝑖 𝑡 with σ t ℝ 𝜎 𝑡 ℝ and σ 1 𝜎 1 then 1 n s 1 n σ 1 n 1 d 1 superscript 𝑛 𝑠 1 superscript 𝑛 𝜎 1 superscript 𝑛 1 𝑑 for a positive d 𝑑 for all n 1 2 𝑛 1 2 the series n 1 1 n 1 d superscript subscript 𝑛 1 1 superscript 𝑛 1 𝑑 converges since 1 d 1 1 𝑑 1 thus the series 1 converges uniformly in the closed half plane ℜ s 1 d 𝑠 1 𝑑 by the Weierstrass criterion Therefore we can infer see theorems on complex function series that the sum ζ s 𝜁 𝑠 of 1 is holomorphic in the domain ℜ s 1 𝑠 1 We use also the fact that the series n 1 1 n 1 n s 1 1 2 s 1 3 s 1 4 s fragments superscript subscript 𝑛 1 superscript

    Original URL path: http://www.planetmath.org/analyticcontinuationofriemannzetatocriticalstrip (2016-04-25)
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  • analytic geometry | planetmath.org
    3E 7B msc 3A30B40 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 3A30B40 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 3A11M41 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 3A11M41 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 analytic geometry Analytic geometry is the branch of geometry that uses mathematical analysis and algebraic calculations for investigating geometric problems Many such problems can be put into the form of equations and by analyzing these equations one may obtain solutions which can be interpreted geometrically The fundamental

    Original URL path: http://www.planetmath.org/analyticgeometry (2016-04-25)
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  • angle between line and plane | planetmath.org
    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 3A01A45 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 3A01A45 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 angle between line and plane The angle between a line l l l and a plane τ τ tau is defined as the least possible angle ω ω omega between l l l and a line contained by τ τ tau It is apparent that ω ω omega satisfies always 0 ω 90 0 ω superscript 90 0 leqq omega leqq 90 circ Let the plane τ τ tau be given by the equation A x B y C z D 0 A x B y C z D 0 Ax By Cz D 0 i e its normal vector has the components A B C A B C A B C Let a direction vector of the line l l l have the components a b c a b c a b c Then the angle ω ω omega between l l l and τ τ tau is obtained from the equation sin ω A a B b C c A 2 B 2 C 2 a 2 b 2 c 2 ω A a B b C c superscript A 2 superscript B 2 superscript C 2 superscript a 2 superscript b 2 superscript c 2 sin omega frac Aa Bb Cc sqrt A 2 B 2 C 2 sqrt a 2 b 2 c 2 In fact the right hand side is the cosine of the angle α α alpha between l l l and the surface normal of τ τ tau see angle between two lines and ω ω

    Original URL path: http://www.planetmath.org/anglebetweenlineandplane (2016-04-25)
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  • angle between two lines | planetmath.org
    if the lines are not parallel If θ θ theta denotes the angle between two lines it always satisfies the inequalities 0 θ π 2 0 θ π 2 displaystyle 0 leqq theta leqq frac pi 2 1 If the slopes of the two lines are m 1 subscript m 1 m 1 and m 2 subscript m 2 m 2 the angle θ θ theta is obtained from tan θ m 1 m 2 1 m 1 m 2 θ subscript m 1 subscript m 2 1 subscript m 1 subscript m 2 displaystyle tan theta left frac m 1 m 2 1 m 1 m 2 right 2 This equation clicks in the case that m 1 m 2 1 subscript m 1 subscript m 2 1 m 1 m 2 1 when the lines are perpendicular and θ θ theta equals to π 2 π 2 displaystyle frac pi 2 Also if one of the lines is parallel to y y y axis it has no slope then the angle θ θ theta must be deduced using the slope of the other line If one of the slopes is 0 0 0 the angle between the two lines is just the angle between one of the lines and the x x x axis Assume the other line has slope m m m then formula 2 above becomes tan θ m θ m displaystyle tan theta left m right 3 If on the other hand one of the slopes is infinite meaning that the line is parallel to the y y y axis then the angle between two lines is the same as the angle between one line with slope m m m and the y y y axis which is tan θ 1 m θ 1 m displaystyle tan theta left frac 1 m right 4 The above formula is consistent with formula 2 in the sense that if we let one of m 1 subscript m 1 m 1 or m 2 subscript m 2 m 2 approach infty we get formula 4 Remark If both slopes are positive then formula 2 above is really just a disguised form of the subtraction formula for tangent psaxes Dx 20 Dy 20 1 0 3 1 6 4 α α alpha β β beta x x x y y y β α β α beta alpha ł 2 subscript ł 2 l 2 ł 1 subscript ł 1 l 1 In the diagram above we see that the angle between the two lines is the algebraic difference of the two angles made between each of the lines and the x x x axis In the Euclidean space the angle θ θ theta between two lines is most comfortably defined by using the direction vectors u normal u vec u and v normal v vec v of the lines cos θ u v u v θ normal normal u normal v normal u normal v cos theta left frac

    Original URL path: http://www.planetmath.org/anglebetweentwolines (2016-04-25)
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  • angle bisector as locus | planetmath.org
    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 angle bisector as locus If 0 α 180 o 0 α superscript 180 normal o 0 alpha 180 mathrm o then the angle bisector of α α alpha is the locus of all such points which are equidistant from both sides of the angle it is proved by using the AAS and SSA theorems The equation of the angle bisectors of all four angles formed by two intersecting lines a 1 x b 1 y c 1 0 a 2 x b 2 y c 2 0 formulae sequence subscript a 1 x subscript b 1 y subscript c 1 0 subscript a 2 x subscript b 2 y subscript c 2 0 displaystyle a 1 x b 1 y c 1 0 qquad a 2 x b 2 y c 2 0 1 is a 1 x b 1 y c 1 a 1 2 b 1 2 a 2 x b 2 y c 2 a 2 2 b 2 2 subscript a 1 x subscript b 1 y subscript c 1 superscript subscript a 1 2 superscript subscript b 1 2 plus or minus subscript a 2 x subscript b 2 y subscript c 2 superscript subscript a 2 2 superscript subscript b 2 2 displaystyle frac a 1 x b 1 y c 1 sqrt a 1 2 b 1 2 pm frac a 2 x b 2 y c 2 sqrt a 2 2 b 2 2 2 which may be written in the form x sin α 1 y cos α 1 h 1 x sin

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