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  • general associativity | planetmath.org
    tab Coauthors PDF Source Edit general associativity If an associative binary operation of a set S S S is denoted by normal cdot the associative law in S S S is usually expressed as a b c a b c normal normal a b c normal a normal b c a cdot b cdot c a cdot b cdot c or leaving out the dots a b c a b c a b c a b c ab c a bc Thus the common value of both sides may be denoted as a b c a b c abc With four elements of S S S we can calculate using only the associativity as follows a b c d a b c d a b c d a b c d a b c d a b c d a b c d a b c d a b c d a b c d ab cd a b cd a bc d a bc d ab c d So we may denote the common value of those five expressions as a b c d a b c d abcd Theorem The expression formed of elements a 1 subscript a 1 a 1 a 2 subscript a 2 a 2 a n subscript a n a n of S S S represents always the same element of S S S independently on how one has joined them together with the associative operation and parentheses if only the order of the elements is every time the same The common value is denoted by a 1 a 2 a n subscript a 1 subscript a 2 normal subscript a n a 1 a 2 ldots a n Note The n n n elements can be joined without changing their order in 2 n 2 n n 1 2 n 2 n n 1 frac 2n 2 n n 1 ways see the Catalan numbers The theorem is proved by induction on n n n The cases n 3 n 3 n 3 and n 4 n 4 n 4 have been stated right above Let n ℤ n subscript ℤ n in mathbb Z The expression a a a a a normal a aa ldots a with n n n equal factors a a a may be denoted by a n superscript a n a n and called a power of a a a If the associative operation is denoted additively then the sum a a a a a normal a a a cdots a of n n n equal elements a a a is denoted by n a n a na and called a multiple of a a a hence in every ring one may consider powers and multiples According to whether n n n is an even or an odd number one may speak of even powers odd powers even multiples odd multiples The following two laws can be proved by induction a m a n a m n normal

    Original URL path: http://www.planetmath.org/generalassociativity (2016-04-25)
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  • general commutativity | planetmath.org
    subscript a 1 subscript a 2 normal subscript a n a 1 a 2 ldots a n in S S S and for each permutation π π pi on 1 2 n 1 2 normal n 1 2 ldots n one has i 1 n a π i i 1 n a i superscript subscript product i 1 n subscript a π i superscript subscript product i 1 n subscript a i displaystyle prod i 1 n a pi i prod i 1 n a i 1 Proof If n 1 n 1 n 1 we have nothing to prove Make the induction hypothesis that 1 is true for n m 1 n m 1 n m 1 Denote π 1 m k i e π k m formulae sequence superscript π 1 m k i e π k m pi 1 m k quad mbox i e quad pi k m Then i 1 m a π i i 1 k 1 a π i a π k i 1 m k a π k i i 1 k 1 a π i i 1 m k a π k i a m superscript subscript product i 1 m subscript a π i superscript subscript product i 1 k 1 normal subscript a π i subscript a π k superscript subscript product i 1 m k subscript a π k i normal superscript subscript product i 1 k 1 normal subscript a π i superscript subscript product i 1 m k subscript a π k i subscript a m prod i 1 m a pi i prod i 1 k 1 a pi i cdot a pi k cdot prod i 1 m k a pi k i left prod i 1 k 1 a pi i cdot prod

    Original URL path: http://www.planetmath.org/generalcommutativity (2016-04-25)
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  • general formulas for integration | planetmath.org
    1 C superscript f x r superscript f normal x d x 1 r 1 superscript f x r 1 C displaystyle int f x r f prime x dx frac 1 r 1 f x r 1 C for r 1 r 1 r neq 1 8 f x f x d x ln f x C superscript f normal x f x d x f x C displaystyle int frac f prime x f x dx ln f x C 9 e f x f x d x e f x C superscript e f x superscript f normal x d x superscript e f x C displaystyle int e f x f prime x dx e f x C 10 f x f x a f x b d x a a b d x f x a b a b d x f x b f x f x a f x b d x a a b d x f x a b a b d x f x b displaystyle int frac f x f x a f x b dx frac a a b int frac dx f x a frac b a b int frac dx f x b 11 sin ω x φ d x cos ω x φ ω C ω x φ d x ω x φ ω C displaystyle int sin omega x varphi dx frac cos omega x varphi omega C 12 cos ω x φ d x sin ω x φ ω C ω x φ d x ω x φ ω C displaystyle int cos omega x varphi dx frac sin omega x varphi omega C 13 sinh ω x φ d x cosh ω x φ ω C ω x φ d x ω x φ ω C displaystyle int sinh omega x varphi dx frac cosh omega x varphi omega C 14 cosh ω x φ d x sinh ω x φ ω C ω x φ d x ω x φ ω C displaystyle int cosh omega x varphi dx frac sinh omega x varphi omega C 15 a x b d x 2 3 a a x b a x b C a x b d x 2 3 a a x b a x b C displaystyle int sqrt ax b dx frac 2 3a ax b sqrt ax b C 16 a x 2 b d x x 2 a x 2 b b 2 a ln x a a x 2 b C a superscript x 2 b d x x 2 a superscript x 2 b b 2 a x a a superscript x 2 b C displaystyle int sqrt ax 2 b dx frac x 2 sqrt ax 2 b frac b 2 sqrt a ln x sqrt a sqrt ax 2 b C 17 sin n x cos m x d x sin n 1 x cos m 1 x m n n 1 m

    Original URL path: http://www.planetmath.org/generalformulasforintegration (2016-04-25)
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  • general power | planetmath.org
    beta ln z alpha arg z This shows that both the modulus and the argument of the general power are in general multivalued The modulus is unique only if β 0 β 0 beta 0 i e if the exponent μ α μ α mu alpha is real in this case we have z μ z μ arg z μ μ arg z formulae sequence superscript z μ superscript z μ superscript z μ normal μ z z mu z mu quad arg z mu mu cdot arg z Let β 0 β 0 beta neq 0 If one lets the point z z z go round the origin anticlockwise arg z z arg z gets an addition 2 π 2 π 2 pi and hence the power z μ superscript z μ z mu has been multiplied by a factor having the modulus e 2 π β 1 superscript e 2 π β 1 e 2 pi beta neq 1 and we may say that z μ superscript z μ z mu has come to a new branch Examples 1 z 1 m superscript z 1 m z frac 1 m where m m m is a positive integer coincides with the m th superscript m th m mathrm th root of z z z 2 3 2 e 2 log 3 e 2 ln 3 2 n π i 9 e 2 π i 2 n 9 superscript 3 2 superscript e 2 3 superscript e 2 3 2 n π i 9 superscript superscript e 2 π i 2 n 9 displaystyle 3 2 e 2 log 3 e 2 ln 3 2n pi i 9 e 2 pi i 2 n 9 n ℤ for all n ℤ forall n in mathbb Z 3 i i e i log i e i ln 1 π 2 i 2 n π i e 2 n π π 2 superscript i i superscript e i i superscript e i 1 π 2 i 2 n π i superscript e 2 n π π 2 displaystyle i i e i log i e i ln 1 frac pi 2 i 2n pi i e 2n pi frac pi 2 with n 0 1 2 n 0 plus or minus 1 plus or minus 2 normal n 0 pm 1 pm 2 ldots all these values are positive real numbers the simplest of them is 1 e π 0 20788 1 superscript e π 0 20788 displaystyle frac 1 sqrt e pi approx 0 20788 4 1 i e 2 n 1 π superscript 1 i superscript e 2 n 1 π 1 i e 2n 1 pi with n 0 1 2 n 0 plus or minus 1 plus or minus 2 normal n 0 pm 1 pm 2 ldots also are situated on the positive real axis 5 1 2 e 2 log 1 e 2 i π 2 n π e i 2 n 1 π 2

    Original URL path: http://www.planetmath.org/generalpower (2016-04-25)
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  • general solution of linear differential equation | planetmath.org
    subscript C 2 superscript e 2 x y prime C 1 prime e 2x C 2 prime e 2x 2C 1 e 2 x 2C 2 e 2x of it reduces to the latter bracket expression if we set the condition C 1 e 2 x C 2 e 2 x 0 superscript subscript C 1 normal superscript e 2 x superscript subscript C 2 normal superscript e 2 x 0 displaystyle C 1 prime e 2x C 2 prime e 2x 0 3 So the second derivative is y 2 C 1 e 2 x 2 C 2 e 2 x 4 C 1 e 2 x 4 C 2 e 2 x superscript y 2 superscript subscript C 1 normal superscript e 2 x 2 superscript subscript C 2 normal superscript e 2 x 4 subscript C 1 superscript e 2 x 4 subscript C 2 superscript e 2 x y prime prime 2C 1 prime e 2x 2C 2 prime e 2x 4C 1 e 2x 4C 2 e 2x Substituting this and the expression of y y y in the differential equation 2 gives the simple equation 2 C 1 e 2 x 2 C 2 e 2 x e x 2 superscript subscript C 1 normal superscript e 2 x 2 superscript subscript C 2 normal superscript e 2 x superscript e x displaystyle 2C 1 prime e 2x 2C 2 prime e 2x e x 4 Now we have the pair of linear equations formed by 3 and 4 for determining the derivatives C 1 superscript subscript C 1 normal C 1 prime and C 2 superscript subscript C 2 normal C 2 prime the result of them is C 1 e x 4 C 2 e 3 x 4 formulae sequence superscript subscript C 1 normal superscript e x 4 superscript subscript C 2 normal superscript e 3 x 4 C 1 prime e x 4 C 2 prime e 3x 4 If we then integrate and chose C 1 e x 4 C 2 e 3 x 12 formulae sequence subscript C 1 superscript e x 4 subscript C 2 superscript e 3 x 12 C 1 e x 4 C 2 e 3x 12 we can form the particular solution y e x 4 e 2 x e 3 x 12 e 2 x e x 3 y superscript e x 4 superscript e 2 x superscript e 3 x 12 superscript e 2 x superscript e x 3 y frac e x 4 e 2x frac e 3x 12 e 2x equiv frac e x 3 Accordingly the general solution of the nonhomogeneous equation 2 is y e x 3 C 1 e 2 x C 2 e 2 x y superscript e x 3 subscript C 1 superscript e 2 x subscript C 2 superscript e 2 x y frac e x 3 C 1 e 2x C 2 e 2x In some cases it is not necessary to use the variation of parameters method above illustrated but a particular solution may be found at simple sight as it is the case in the following example about boundary values Example 2 Find the general solution of the nonhomogeneous linear second order differential equation y y 2 x superscript y y 2 x y prime prime y 2x under the boundary conditions y 1 0 y 0 0 formulae sequence y 1 0 superscript y normal 0 0 y 1 0 quad y prime 0 0 The function x 2 x maps to x 2 x x mapsto 2x is evidently a particular solution of the differential equation Therefore the general solution is y x 2 x C 1 e x C 2 e x y x 2 x subscript C 1 superscript e x subscript C 2 superscript e x y x 2x C 1 e x C 2 e x Thus we have y x 2 C 1 e x C 2 e x superscript y normal x 2 subscript C 1 superscript e x subscript C 2 superscript e x y prime x 2 C 1 e x C 2 e x By making use of the boundary conditions we obtain 0 y 1 2 C 1 e C 2 e 1 0 y 0 2 C 1 C 2 formulae sequence 0 y 1 2 subscript C 1 e subscript C 2 superscript e 1 0 superscript y normal 0 2 subscript C 1 subscript C 2 0 y 1 2 C 1 e C 2 e 1 qquad 0 y prime 0 2 C 1 C 2 Solving this system of linear equations and introducing C 1 subscript C 1 C 1 and C 2 subscript C 2 C 2 into the general solution we have the result y x 2 x 2 e 1 e 2 1 e x 2 e e 1 e 2 1 e x y x 2 x 2 e 1 superscript e 2 1 superscript e x 2 e e 1 superscript e 2 1 superscript e x y x 2x frac 2 e 1 e 2 1 e x frac 2e e 1 e 2 1 e x To solve more advanced problems about nonhomogeneous ordinary linear differential equations of second order with boundary conditions we may find out a particular solution by using for instance the Green s function method Thus consider for instance the self adjoint differential equation 1 1 Minus sign on the right hand member of the equation it is by convenience in the applications d d x p x d y d x q x y f x a x b y a y b 0 formulae sequence formulae sequence d d x p x d y d x q x y f x a x b y a y b 0 frac d dx left p x frac dy dx right q x y f x qquad a x b qquad y

    Original URL path: http://www.planetmath.org/generalsolutionoflineardifferentialequation (2016-04-25)
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  • generalisation of Gaussian integral | planetmath.org
    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 3A34A05 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 3A34A05 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 generalisation of Gaussian integral The integral 0 e x 2 cos t x d x w t assign superscript subscript 0 superscript e superscript x 2 t x d x w t int 0 infty e x 2 cos tx dx w t is a generalisation of the Gaussian integral w 0 π 2 w 0 π 2 w 0 frac sqrt pi 2 For evaluating it we first form its derivative which may be done by differentiating under the integral sign w t 0 e x 2 x sin t x d x 1 2 0 e x 2 2 x sin t x d x superscript w normal t superscript subscript 0 superscript e superscript x 2 x t x d x 1 2 superscript subscript 0 superscript e superscript x 2 2 x t x d x w prime t int 0 infty e x 2 x sin tx dx frac 1 2 int 0 infty e x 2 2x sin tx dx Using integration by parts this yields w t 1 2 x 0 e x 2 sin t x t 2 0 e x 2 cos t x d x 1 2 0 0 t 2 0 e x 2 cos t x d x t 2 w t superscript w normal t 1 2 superscript subscript normal x 0 superscript e superscript x 2 t x t 2 superscript subscript 0 superscript e superscript x 2 t x d x 1 2 0 0 t 2 superscript subscript 0 superscript e superscript x 2 t x d x t 2 w t w prime t frac 1 2

    Original URL path: http://www.planetmath.org/generalisationofgaussianintegral (2016-04-25)
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  • generalized binomial coefficients | planetmath.org
    2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A36 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 3A26A36 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 generalized binomial coefficients The binomial coefficients n r n n r r binomial n r n n r r displaystyle n choose r frac n n r r 1 where n n n is a non negative integer and r 0 1 2 n r 0 1 2 normal n r in 0 1 2 ldots n can be generalized for all integer and non integer values of n n n by using the reduced form n r n n 1 n 2 n r 1 r binomial n r n n 1 n 2 normal n r 1 r displaystyle n choose r frac n n 1 n 2 ldots n r 1 r 2 here r r r may be any non negative integer Then Newton s binomial series gets the simple form 1 z α r 0 α r z r 1 α 1 z α 2 z 2 superscript 1 z α superscript subscript r 0 binomial α r superscript z r 1 binomial α 1 z binomial α 2 superscript z 2 normal displaystyle 1 z alpha sum r 0 infty alpha choose r z r 1 alpha choose 1 z alpha choose 2 z 2 cdots 3 It is not hard to show that the radius of convergence of this series is 1 This series expansion is valid for every complex number α α alpha when z 1 z 1 z 1 and it presents such a branch of the power 1 z α superscript 1 z α 1 z alpha which gets the value 1 in the point z 0 z 0 z 0 In the case that α α alpha is a non negative integer and r r r is great enough one factor in the numerator of α r α α

    Original URL path: http://www.planetmath.org/generalizedbinomialcoefficients (2016-04-25)
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  • generating function of Hermite polynomials | 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 3A05A10 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 3A05A10 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 generating function of Hermite polynomials We start from the definition of Hermite polynomials via their Rodrigues formula H n z 1 n e z 2 d n d z n e z 2 n 0 1 2 fragments subscript H n fragments normal z normal assign superscript fragments normal 1 normal n superscript e superscript z 2 superscript d n d superscript z n superscript e superscript z 2 italic fragments normal n 0 normal 1 normal 2 normal normal normal normal displaystyle H n z 1 n e z 2 frac d n dz n e z 2 qquad n 0 1 2 ldots 1 The consequence f n z n 2 π i C f ζ ζ z n 1 d ζ superscript f n z n 2 π i subscript contour integral C f ζ superscript ζ z n 1 d ζ displaystyle f n z frac n 2 pi i oint C frac f zeta zeta z n 1 d zeta 2 of Cauchy integral formula allows to write 1 as the complex integral H n z 1 n n 2 i π C e z 2 ζ 2 ζ z n 1 d ζ subscript H n z superscript 1 n n 2 i π subscript contour integral C superscript e superscript z 2 superscript ζ 2 superscript ζ z n 1 d ζ H n z 1 n frac n 2i pi oint C frac e z 2 zeta 2 zeta z n 1 d zeta where C C C is any closed contour around the point z z z and the direction is anticlockwise The substitution z ζ t assign z ζ

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