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  • any nonzero integer is quadratic residue | planetmath.org
    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 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 3A12J20 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 3A12J20 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 any nonzero integer is quadratic residue Theorem For every nonzero integer a a a there exists an odd prime number p p p such that a a a is a quadratic residue modulo p p p Proof 1 superscript 1 1 circ a 2 a 2 a 2 We see that 3 2 2 mod 7 superscript 3 2 annotated 2 pmod 7 3 2 equiv 2 mathop rm mod 7 and 7 2 not divides 7 2 7 nmid 2 whence 2 is a quadratic residue

    Original URL path: http://www.planetmath.org/anynonzerointegerisquadraticresidue (2016-04-25)
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  • Appell sequence | planetmath.org
    displaystyle langle P 0 x P 1 x P 2 x ldots rangle 1 with P n x a x n n 0 1 2 fragments subscript P n fragments normal x normal assign a superscript x n italic fragments normal n 0 normal 1 normal 2 normal normal normal P n x ax n qquad n 0 1 2 ldots is a geometric sequence and has trivially the properties P n x n P n 1 x n 0 1 2 fragments superscript subscript P n normal fragments normal x normal n subscript P n 1 fragments normal x normal italic fragments normal n 0 normal 1 normal 2 normal normal normal displaystyle P n prime x nP n 1 x qquad n 0 1 2 ldots 2 and P n x y k 0 n n k P k x y n k subscript P n x y superscript subscript k 0 n binomial n k subscript P k x superscript y n k displaystyle P n x y sum k 0 n n choose k P k x y n k 3 see the binomial theorem There are also other polynomial sequences 1 having these properties for example the sequences of the Bernoulli polynomials the Euler polynomials and the Hermite polynomials Such sequences are called Appell sequences and their members are sometimes characterised as generalised monomials because of resemblance to the geometric sequence Given the first member P 0 x subscript P 0 x P 0 x which must be a nonzero constant polynomial of any Appell sequence 1 the other members are determined recursively by P n x 0 x P n 1 t d t C n subscript P n x superscript subscript 0 x subscript P n 1 t d t subscript C n displaystyle P n x int 0 x P n 1 t dt C n 4 as one gives the values of the constants of integration C n subscript C n C n thus the number sequence C 0 C 1 C 2 subscript C 0 subscript C 1 subscript C 2 normal langle C 0 C 1 C 2 ldots rangle determines the Appell sequence uniquely So the choice C 1 C 2 0 subscript C 1 subscript C 2 normal assign 0 C 1 C 2 ldots 0 yields a geometric sequence and the choice C n B n assign subscript C n subscript B n C n B n for n 0 1 2 n 0 1 2 normal n 0 1 2 ldots the Bernoulli polynomials The properties 2 and 3 are equivalent The implication 2 3 normal 2 3 2 Rightarrow 3 may be shown by induction on n n n The reverse implication is gotten by using the definition of derivative P n x superscript subscript P n normal x displaystyle P n prime x lim Δ x 0 P n x Δ x P n x Δ x normal subscript normal normal Δ x

    Original URL path: http://www.planetmath.org/appellsequence (2016-04-25)
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  • application of Cauchy criterion for convergence | planetmath.org
    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 3A11C08 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 3A11B83 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 3A11B83 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

    Original URL path: http://www.planetmath.org/applicationofcauchycriterionforconvergence (2016-04-25)
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  • application of Cauchy--Schwarz inequality | planetmath.org
    Coauthors PDF Source Edit application of Cauchy Schwarz inequality In determining the perimetre of ellipse one encounters the elliptic integral 0 π 2 1 ε 2 sin 2 t d t superscript subscript 0 π 2 1 superscript ε 2 superscript 2 t d t int 0 frac pi 2 sqrt 1 varepsilon 2 sin 2 t dt where the parametre ε ε varepsilon is the eccentricity of the ellipse 0 ε 1 0 ε 1 0 leqq varepsilon 1 A good upper bound for the integral is obtained by utilising the Cauchy Schwarz inequality a b f g a b f 2 a b g 2 superscript subscript a b f g superscript subscript a b superscript f 2 superscript subscript a b superscript g 2 left int a b fg right leqq sqrt int a b f 2 sqrt int a b g 2 choosing in it f t 1 assign f t 1 f t 1 and g t 1 ε 2 sin 2 t assign g t 1 superscript ε 2 superscript 2 t g t sqrt 1 varepsilon 2 sin 2 t Then we get 0 0 π 2 1 ε 2 sin 2 t d t 0 superscript subscript 0 π 2 1 superscript ε 2 superscript 2 t d t displaystyle 0 int 0 frac pi 2 sqrt 1 varepsilon 2 sin 2 t dt 0 π 2 1 2 d t 0 π 2 1 ε 2 sin 2 t d t absent superscript subscript 0 π 2 superscript 1 2 d t superscript subscript 0 π 2 1 superscript ε 2 superscript 2 t d t displaystyle leqq sqrt int 0 frac pi 2 1 2 dt sqrt int 0 frac pi 2 left 1 varepsilon 2 sin 2 t right

    Original URL path: http://www.planetmath.org/applicationofcauchyschwarzinequality (2016-04-25)
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  • application of fundamental theorem of integral calculus | planetmath.org
    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 Primary tabs View active tab Coauthors PDF Source Edit application of fundamental theorem of integral calculus We will derive the addition formulas of the sine and the cosine functions supposing known only their derivatives and the chain rule Define the function F ℝ ℝ normal F normal ℝ ℝ F mathbb R to mathbb R through F x sin x cos α cos x sin α sin x α 2 cos x cos α sin x sin α cos x α 2 assign F x superscript x α x α x α 2 superscript x α x α x α 2 F x sin x cos alpha cos x sin alpha sin x alpha 2 cos x cos alpha sin x sin alpha cos x alpha 2 where α α alpha is for the moment a constant The derivative of F F F is easily calculated F x 2 sin x cos α cos x sin α sin x α cos x cos α sin x sin α cos x α 2 cos x cos α sin x sin α cos x α sin x cos α cos x sin α sin x α superscript F normal x 2 x α x α x α x α x α x α 2 x α x α x α x α x α x α F prime x mbox 2 sin x cos alpha cos x sin alpha sin x alpha cos x cos alpha sin x sin alpha cos x alpha 2 cos x cos alpha sin x sin alpha cos x alpha sin x cos

    Original URL path: http://www.planetmath.org/applicationoffundamentaltheoremofintegralcalculus (2016-04-25)
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  • application of logarithm series | planetmath.org
    92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit application of logarithm series The integrand of the improper integral I 0 1 ln 1 x x d x assign I superscript subscript 0 1 1 x x d x displaystyle I int 0 1 frac ln 1 x x dx 1 is not defined at the lower limit 0 However from the Taylor series expansion ln 1 x x x 2 2 x 3 3 x 4 4 1 x 1 fragments fragments normal 1 x normal x superscript x 2 2 superscript x 3 3 superscript x 4 4 normal italic fragments normal 1 x 1 normal ln 1 x x frac x 2 2 frac x 3 3 frac x 4 4 ldots qquad 1 x leqq 1 of the natural logarithm we obtain the expansion of the integrand ln 1 x x 1 x 2 x 2 3 x 3 4 1 x 0 0 x 1 fragments 1 x x 1 x 2 superscript x 2 3 superscript x 3 4 normal italic fragments normal 1 x 0 normal 0 x 1 normal frac ln 1 x x 1 frac x 2 frac x 2 3 frac x 3 4 ldots qquad 1 x 0 0 x leqq 1 whence lim x 0 ln 1 x x 1 subscript normal x 0 1 x x 1 displaystyle lim x to 0 frac ln 1 x x 1 2 This implies that the integrand of 1 is bounded on the interval 0 1 0 1 0 1 and also continuous if we think that 2 defines its value at x 0 x 0 x 0 Accordingly the integrand is Riemann integrable on the interval and we can determine the improper integral by integrating termwise I I displaystyle I 0 1 1 x 2 x 2 3 x 3 4 d x fragments superscript subscript 0 1 fragments normal 1 x 2 superscript x 2 3 superscript x 3 4 normal normal d x displaystyle int 0 1 left 1 frac x 2 frac x 2 3 frac x 3 4 ldots right dx 0 1 x x 2 2 2 x 3 3 2 x 4 4 2 fragments superscript subscript normal 0 1 fragments normal x superscript x 2 superscript 2 2 superscript x 3 superscript 3 2 superscript x 4 superscript 4 2 normal normal displaystyle operatornamewithlimits Big 0 quad 1 left x frac x 2 2 2 frac x 3 3 2 frac x 4 4 2 ldots right 1 1 2 2 1 3 2 1 4 2 fragments 1 1 superscript 2 2 1 superscript 3 2 1 superscript 4 2 normal displaystyle 1 frac 1 2 2 frac 1 3 2 frac 1 4 2 ldots By the entry on Dirichlet eta function at 2 the sum of the obtained series is η 2 π 2 12 η 2 superscript

    Original URL path: http://www.planetmath.org/applicationoflogarithmseries (2016-04-25)
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  • application of sine integral at infinity | planetmath.org
    23this 3E 7B msc 3A33B10 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 application of sine integral at infinity For finding the value of the improper integral 0 sin a x x 1 x 2 d x f a a 0 fragments superscript subscript 0 a x x 1 superscript x 2 d x assign f fragments normal a normal italic fragments normal a 0 normal displaystyle int 0 infty frac sin ax x 1 x 2 dx f a qquad a 0 1 we first use the partial fraction representation 1 x 1 x 2 1 x x 1 x 2 1 x 1 superscript x 2 1 x x 1 superscript x 2 frac 1 x 1 x 2 frac 1 x frac x 1 x 2 Thus we may write f a 0 sin a x x d x 0 x sin a x 1 x 2 d x f a superscript subscript 0 a x x d x superscript subscript 0 x a x 1 superscript x 2 d x f a int 0 infty frac sin ax x dx int 0 infty frac x sin ax 1 x 2 dx But by the entry sine integral at infinity the first integral equals π 2 π 2 displaystyle frac pi 2 When we check f a 0 cos a x 1 x 2 d x f a 0 x sin a x 1 x 2 d x formulae sequence superscript f normal a superscript subscript 0 a x 1 superscript x 2 d x superscript f a superscript subscript 0 x a x 1 superscript x 2 d x f prime a int 0 infty frac cos ax 1 x 2 dx quad f prime prime a int 0 infty frac x sin ax 1 x 2 dx we see that there is the linear differential equation f a π 2 f a f a π 2 superscript f a displaystyle f a frac pi 2 f prime prime a 2 i e f f π 2 superscript f f π 2 f prime prime f frac pi 2 satisfied by the sought function a f a maps to a f a a mapsto f a We have the initial conditions f 0 0 0 d x 0 f 0 0 d x 1 x 2 0 arctan x π 2 formulae sequence f 0 superscript subscript 0 0 d x 0 superscript f normal 0 superscript subscript 0 d x 1 superscript x 2 superscript subscript normal 0 x π 2 f 0 int 0 infty 0 dx 0 quad f prime 0 int 0 infty frac dx 1 x 2 operatornamewithlimits Big 0 quad infty arctan x frac pi 2 Therefore the general solution f a C 1 e

    Original URL path: http://www.planetmath.org/applicationofsineintegralatinfinity (2016-04-25)
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  • applying elementary symmetric polynomials | planetmath.org
    2 normal superscript subscript x n subscript λ n x 1 lambda 1 x 2 lambda 2 cdots x n lambda n where λ 1 λ 2 λ n 0 and λ 1 λ 2 λ n d formulae sequence subscript λ 1 subscript λ 2 normal subscript λ n 0 and subscript λ 1 subscript λ 2 normal subscript λ n d lambda 1 geq lambda 2 geq ldots geq lambda n geq 0 quad mbox and quad lambda 1 lambda 2 ldots lambda n d Then P Q p 1 p 2 p n i m i p 1 λ 1 λ 2 p 2 λ 2 λ 3 p n 1 λ n 1 λ n p n λ n P Q subscript p 1 subscript p 2 normal subscript p n subscript i subscript m i superscript subscript p 1 subscript λ 1 subscript λ 2 superscript subscript p 2 subscript λ 2 subscript λ 3 normal superscript subscript p n 1 subscript λ n 1 subscript λ n superscript subscript p n subscript λ n displaystyle P Q p 1 p 2 ldots p n sum i m i p 1 lambda 1 lambda 2 p 2 lambda 2 lambda 3 cdots p n 1 lambda n 1 lambda n p n lambda n 1 in which the coefficients m i subscript m i m i are determined by giving some suitable values to the indeterminates x j subscript x j x j Example 1 Express the polynomial P x 1 3 x 2 x 1 3 x 3 x 2 3 x 1 x 2 3 x 3 x 3 3 x 1 x 3 3 x 2 P superscript subscript x 1 3 subscript x 2 superscript subscript x 1 3 subscript x 3 superscript subscript x 2 3 subscript x 1 superscript subscript x 2 3 subscript x 3 superscript subscript x 3 3 subscript x 1 superscript subscript x 3 3 subscript x 2 P x 1 3 x 2 x 1 3 x 3 x 2 3 x 1 x 2 3 x 3 x 3 3 x 1 x 3 3 x 2 in the elementary symmetric polynomials p 1 x 1 x 2 x 3 p 2 x 2 x 3 x 3 x 1 x 1 x 2 p 3 x 1 x 2 x 3 formulae sequence subscript p 1 subscript x 1 subscript x 2 subscript x 3 formulae sequence subscript p 2 subscript x 2 subscript x 3 subscript x 3 subscript x 1 subscript x 1 subscript x 2 subscript p 3 subscript x 1 subscript x 2 subscript x 3 displaystyle p 1 x 1 x 2 x 3 quad p 2 x 2 x 3 x 3 x 1 x 1 x 2 quad p 3 x 1 x 2 x 3 2 We have four exponent combinations 4 0 0 3 1 0 2 2 0 2 1 1 4 0 0 3 1 0 2 2 0 2 1 1 4 0 0 quad 3 1 0 quad 2 2 0 quad 2 1 1 for which the corresponding p p p products of the sum 1 are p 1 4 p 1 2 p 2 p 2 2 p 1 p 3 superscript subscript p 1 4 superscript subscript p 1 2 subscript p 2 superscript subscript p 2 2 subscript p 1 subscript p 3 p 1 4 quad p 1 2 p 2 quad p 2 2 quad p 1 p 3 respectively Apparently the first one is out of the question Therefore clearly P p 1 2 p 2 a p 2 2 b p 1 p 3 P superscript subscript p 1 2 subscript p 2 a superscript subscript p 2 2 b subscript p 1 subscript p 3 P p 1 2 p 2 ap 2 2 bp 1 p 3 Using x 1 x 2 1 subscript x 1 subscript x 2 1 x 1 x 2 1 and x 3 0 subscript x 3 0 x 3 0 makes p 1 2 subscript p 1 2 p 1 2 p 2 1 subscript p 2 1 p 2 1 and p 3 0 subscript p 3 0 p 3 0 when P 2 4 a 0 P 2 4 a 0 P 2 4 a 0 implying a 2 a 2 a 2 Using similarly x 1 x 2 x 3 1 subscript x 1 subscript x 2 subscript x 3 1 x 1 x 2 x 3 1 we get p 1 p 2 3 subscript p 1 subscript p 2 3 p 1 p 2 3 p 3 1 subscript p 3 1 p 3 1 which give P 6 27 9 a 3 b 9 3 b P 6 27 9 a 3 b 9 3 b P 6 27 9a 3b 9 3b yielding b 1 b 1 b 1 Hence we have the result P p 1 2 p 2 2 p 2 2 p 1 p 3 P superscript subscript p 1 2 subscript p 2 2 superscript subscript p 2 2 subscript p 1 subscript p 3 P p 1 2 p 2 2p 2 2 p 1 p 3 i e x 1 3 x 2 x 1 3 x 3 x 2 3 x 1 x 2 3 x 3 x 3 3 x 1 x 3 3 x 2 x 1 x 2 x 3 2 x 2 x 3 x 3 x 1 x 1 x 2 2 x 2 x 3 x 3 x 1 x 1 x 2 2 x 1 x 2 x 3 x 1 x 2 x 3 superscript subscript x 1 3 subscript x 2 superscript subscript x 1 3 subscript x 3 superscript subscript x 2 3 subscript x 1 superscript subscript x 2 3 subscript x 3 superscript subscript x 3 3

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