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  • complementary angles | planetmath.org
    2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51 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 3A51M15 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 3A51M15 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

    Original URL path: http://www.planetmath.org/complementaryangles (2016-04-25)
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  • complete ultrametric field | planetmath.org
    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 3A51F20 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 3A51F20 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 complete ultrametric field A field K K K equipped with a non archimedean valuation fragments normal normal normal cdot is called a non archimedean field or also an ultrametric field since the valuation induces the ultrametric d x y x y assign d x y x y d x y x y of K K K Theorem Let K d K d K d be a complete ultrametric field A necessary and sufficient condition for the convergence of the series a 1 a 2 a 3 subscript a 1 subscript a 2 subscript a 3 normal displaystyle a 1 a 2 a 3 ldots 1 in K K K is that lim n a n 0 subscript normal n subscript a n 0 displaystyle lim n to infty a n 0 2 Proof Let ε ε varepsilon be any positive number When 1 converges it satisfies the Cauchy condition and therefore exists a number m ε subscript m ε m varepsilon such that surely a m 1 j 1 m 1 a j j 1 m a j ε subscript a m 1 superscript subscript j 1 m 1 subscript a j superscript subscript j 1 m subscript a j ε a m 1 left sum j 1 m 1 a j sum j 1 m a j right varepsilon for all m m ε m subscript m ε m geqq m

    Original URL path: http://www.planetmath.org/completeultrametricfield (2016-04-25)
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  • complex exponential function | planetmath.org
    2Frdf sink 23this 3E 7B msc 3A54E35 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 3A54E35 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 complex exponential function The complex exponential function exp ℂ ℂ normal normal ℂ ℂ exp mathbb C to mathbb C may be defined in many equivalent ways Let z x i y z x i y z x iy where x y ℝ x y ℝ x y in mathbb R exp z e x cos y i sin y assign z superscript e x y i y displaystyle exp z e x cos y i sin y exp z lim n 1 z n n assign z subscript normal n superscript 1 z n n displaystyle exp z lim n to infty left 1 frac z n right n exp z n 0 z n n assign z superscript subscript n 0 superscript z n n displaystyle exp z sum n 0 infty frac z n n The complex exponential function is usually denoted in power form e z exp z assign superscript e z z e z exp z where e e e is the Napier s constant It also coincincides with the real exponential function when z z z is real choose y 0 y 0 y 0 It has all the properties of power e g e z 1 e z superscript e z 1 superscript e z e z frac 1 e z these are consequences of the addition formula e z 1 z 2 e z 1 e z 2 superscript e subscript z 1 subscript z 2 superscript e subscript z 1 superscript e subscript z 2 e z 1 z 2 e z 1 e z 2 of the complex exponential function The function gets all complex values except 0 and is periodic having the prime period the period with least non zero modulus 2 π i 2 π i 2 pi i The exp exp is holomorphic

    Original URL path: http://www.planetmath.org/complexexponentialfunction (2016-04-25)
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  • complex logarithm | planetmath.org
    msc 3A30D20 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 3A32A05 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 3A32A05 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 complex logarithm The logarithm of a complex number z z z is defined as every complex number w w w which satisfies the equation e w z superscript e w z displaystyle e w z 1 This is is denoted by log z w assign z w log z w The solution of 1 is obtained by using the form e w r e i φ superscript e w r superscript e i φ e w re i varphi where r z r z r z and φ arg z φ z varphi arg z the result is w log z ln z i arg z w z z i z w log z ln z i arg z Here the ln z z ln z means the usual Napierian or natural logarithm logarithmus naturalis of the real number z z z If we fix the phase angle φ φ varphi of z z z so that 0 φ 2 π 0 φ 2 π 0 leqq varphi 2 pi we can write log z ln r i φ n 2 π i n 0 1 2 fragments z r i φ n

    Original URL path: http://www.planetmath.org/complexlogarithm (2016-04-25)
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  • complex sine and cosine | planetmath.org
    power series too Thus they define holomorphic functions in the whole complex plane i e entire functions to be more precise entire transcendental functions The series also show that sine is an odd function and cosine an even function Expanding the complex exponential functions e i z superscript e i z e iz and e i z superscript e i z e iz to power series and separating the terms of even and odd degrees gives the generalized Euler s formulas e i z cos z i sin z e i z cos z i sin z formulae sequence superscript e i z z i z superscript e i z z i z e iz cos z i sin z quad e iz cos z i sin z Adding subtracting and multiplying these two formulae give respectively the two Euler s formulae cos z e i z e i z 2 sin z e i z e i z 2 i formulae sequence z superscript e i z superscript e i z 2 z superscript e i z superscript e i z 2 i displaystyle cos z frac e iz e iz 2 quad sin z frac e iz e iz 2i 1 which sometimes are used to define cosine and sine and the fundamental formula of trigonometry cos 2 z sin 2 z 1 superscript 2 z superscript 2 z 1 cos 2 z sin 2 z 1 As consequences of the generalized Euler s formulae one gets easily the addition formulae of sine and cosine sin z 1 z 2 sin z 1 cos z 2 cos z 1 sin z 2 subscript z 1 subscript z 2 subscript z 1 subscript z 2 subscript z 1 subscript z 2 sin z 1 z 2 sin z 1 cos z 2 cos z 1 sin z 2 cos z 1 z 2 cos z 1 cos z 2 sin z 1 sin z 2 subscript z 1 subscript z 2 subscript z 1 subscript z 2 subscript z 1 subscript z 2 cos z 1 z 2 cos z 1 cos z 2 sin z 1 sin z 2 so they are in ℂ ℂ mathbb C fully similar as in ℝ ℝ mathbb R It means that all goniometric formulae derived from these such as sin 2 z 2 sin z cos z sin π z sin z sin 2 z 1 cos 2 z 2 formulae sequence 2 z 2 z z formulae sequence π z z superscript 2 z 1 2 z 2 sin 2z 2 sin z cos z quad sin pi z sin z quad sin 2 z frac 1 cos 2z 2 have the old shape See also the persistence of analytic relations The addition formulae may be written also as sin x i y sin x cosh y i cos x sinh y x i y x y i x y sin x iy sin x cosh y i cos x

    Original URL path: http://www.planetmath.org/complexsineandcosine (2016-04-25)
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  • complex tangent and cotangent | planetmath.org
    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 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 complex tangent and cotangent The tangent and the cotangent function for complex values of the argument z z z are defined with the equations tan z sin z cos z cot z cos z sin z formulae sequence assign z z z assign z z z tan z frac sin z cos z quad cot z frac cos z sin z Using the Euler s formulae one also can define tan z i e i z e i z e i z e i z cot z i e i z e i z e i z e i z formulae sequence assign z i superscript e i z superscript e i z superscript e i z superscript e i z assign z i superscript e i z superscript e i z superscript e i z superscript e i z displaystyle tan z i frac e iz e iz e iz e iz quad cot z i frac e iz e iz e iz e iz 1 The subtraction formulae of cosine and sine yield an additional connection between the cotangent and tangent cot π 2 z cos π 2 z sin π 2 z cos π 2 cos z sin π 2 sin z sin π 2 cos z cos π 2 sin z sin z cos z tan z π 2 z π 2 z π 2 z π 2 z π 2 z π 2 z π 2 z z z z cot frac pi 2 z frac cos frac pi 2 z sin frac pi 2 z frac cos frac pi 2 cos z sin frac pi 2 sin z sin frac pi 2 cos z cos frac pi 2 sin z frac sin z cos z tan z Thus the properties of the tangent are easily derived from the corresponding properties of the cotangent Because of the identic equation cos 2 z sin 2 z 1 superscript 2 z superscript 2 z 1 cos 2 z sin 2 z 1

    Original URL path: http://www.planetmath.org/complextangentandcotangent (2016-04-25)
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  • condition for power basis | planetmath.org
    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 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 condition for power basis Lemma If K K K is an algebraic number field of degree n n n and the elements α 1 α 2 α n subscript α 1 subscript α 2 normal subscript α n alpha 1 alpha 2 ldots alpha n of K K K can be expressed as linear combinations α 1 c 11 β 1 c 12 β 2 c 1 n β n α 2 c 21 β 1 c 22 β 2 c 2 n β n α n c n 1 β 1 c n 2 β 2 c n n β n α1 c11β1 c12β2 c 1nβn α2 c21β1 c22β2 c 2nβn αn c n1β1 c n2β2 c nnβn displaystyle begin cases alpha 1 c 11 beta 1 c 12 beta 2 ldots c 1n beta n alpha 2 c 21 beta 1 c 22 beta 2 ldots c 2n beta n cdots alpha n c n1 beta 1 c n2 beta 2 ldots c nn beta n end cases of the elements β 1 β 2 β n subscript β 1 subscript β 2 normal subscript β n beta 1 beta 2 ldots beta n of K K K with rational coefficients c i j subscript c i j c ij then the discriminants of α i subscript α i alpha i and β j subscript β j beta j are related by the equation Δ α 1 α 2 α n det c i j 2 Δ β 1 β 2 β n normal Δ subscript α 1 subscript α 2 normal subscript α n normal superscript subscript c i j 2 normal Δ subscript β 1 subscript β 2 normal subscript β n Delta alpha 1 alpha 2 ldots alpha n det c ij 2 cdot Delta beta 1 beta 2 ldots beta n Theorem Let ϑ ϑ vartheta be an algebraic integer of degree n n n The set 1 ϑ ϑ n 1 1 ϑ normal superscript ϑ n 1 1 vartheta ldots vartheta n 1 is an integral basis of ℚ

    Original URL path: http://www.planetmath.org/conditionforpowerbasis (2016-04-25)
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  • condition of orthogonality | 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 3A11R04 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 condition of orthogonality Let two straight lines of the x y x y xy plane have the slopes m 1 subscript m 1 m 1 and m 2 subscript m 2 m 2 The lines are at right angles to each other iff m 1 subscript m 1 m 1 and m 2 subscript m 2 m 2 are the opposite inverses of each other i e iff m 1 m 2 1 subscript m 1 subscript m 2 1 m 1 m 2 1 Example The lines y 1 2 x y 1 2 x y 1 sqrt 2 x and y 1 2 x y 1 2 x y 1 sqrt 2 x are at right angles to each other Defines opposite inverse Keywords straight line slope Related OrthogonalCurves InverseNumber OppositeNumber NormalLine AngleBetweenTwoLines PerpendicularityInEuclideanPlane

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