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  • applying generating function | planetmath.org
    e 2 z t superscript t 2 superscript subscript m 1 subscript H m z m 1 superscript t m 1 2z 2t e 2zt t 2 sum m 1 infty frac H m z m 1 t m 1 Here we substitute 1 to the left hand side and rewrite the right hand side getting n 0 2 z H n z n t n n 1 2 H n 1 z n 1 t n n 0 H n 1 z n t n superscript subscript n 0 2 z subscript H n z n superscript t n superscript subscript n 1 2 subscript H n 1 z n 1 superscript t n superscript subscript n 0 subscript H n 1 z n superscript t n sum n 0 infty frac 2zH n z n t n sum n 1 infty frac 2H n 1 z n 1 t n sum n 0 infty frac H n 1 z n t n where we can compare the coefficients of t n superscript t n t n 2 z H n n 2 H n 1 n 1 H n 1 n n 1 2 fragments 2 z subscript H n n 2 subscript H n 1 n 1 subscript H n 1 n italic fragments normal n 1 normal 2 normal normal normal frac 2zH n n frac 2H n 1 n 1 frac H n 1 n qquad n 1 2 ldots Thus we have gotten the recurrence relation H n 1 z 2 z H n z 2 n H n 1 z n 1 2 fragments subscript H n 1 fragments normal z normal 2 z subscript H n fragments normal z normal 2 n subscript H n 1 fragments normal z normal italic fragments normal n 1 normal 2 normal normal normal normal displaystyle H n 1 z 2zH n z 2nH n 1 z qquad n 1 2 ldots 2 Differentiating 1 partially with respect to z z z enables respectively to find a formula expressing the derivative H n z superscript subscript H n normal z H n prime z via the polynomials themselves 2 We copy the equation 1 twice in the forms n 0 H n x n t n e 2 x t t 2 n 0 H n x n u n e 2 x u u 2 formulae sequence superscript subscript n 0 subscript H n x n superscript t n superscript e 2 x t superscript t 2 superscript subscript n 0 subscript H n x n superscript u n superscript e 2 x u superscript u 2 sum n 0 infty frac H n x n t n e 2xt t 2 quad sum n 0 infty frac H n x n u n e 2xu u 2 multiply these with each other and by e x 2 superscript e superscript x 2 e x 2 and then integrate the obtained equation termwise over ℝ

    Original URL path: http://www.planetmath.org/applyinggeneratingfunction (2016-04-25)
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  • arbelos and parbelos | planetmath.org
    2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26C05 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 3A26C05 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 3A26A42 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 3A26A42 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 pspicture 7 1 7 7 psarc linecolor red 0 0 60180 psarc linecolor red 2 25 0 3 750180 psarc linecolor red 3 75 0 2 250180 rput 1 4 An arbelos rput 6 5 0 5 rput 6 5 6

    Original URL path: http://www.planetmath.org/arbelosandparbelos (2016-04-25)
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  • arc length example | planetmath.org
    2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A01A20 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 3A01A20 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 3A00A08 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 3A00A08 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 arc length example The functions x ln sin x and x ln cos x formulae sequence maps to x x and maps to x x x mapsto ln sin x quad mbox and quad x mapsto ln cos x

    Original URL path: http://www.planetmath.org/arclengthexample (2016-04-25)
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  • arc length of logarithmic curve | planetmath.org
    expression s a b 1 x 2 x d x s superscript subscript a b 1 superscript x 2 x d x displaystyle s int a b frac sqrt 1 x 2 x dx 1 Here finding a suitable substitution for integration may be a bit difficult E g x tan t assign x t x tan t leads to 1 x 2 x d x d t sin t cos 2 t 1 superscript x 2 x d x d t t superscript 2 t int frac sqrt 1 x 2 x dx int frac dt sin t cos 2 t the substitution x sinh t assign x t x sinh t to 1 x 2 x d x cosh 2 t sinh t d t 1 superscript x 2 x d x superscript 2 t t d t int frac sqrt 1 x 2 x dx int frac cosh 2 t sinh t dt which both seem to require a new substitution As well the Euler s substitutions 1st and 2nd ones lead to awkward rational functions as integrands But there is the straightforward simple substitution 1 x 2 t x t 2 1 d x t d t t 2 1 formulae sequence assign 1 superscript x 2 t formulae sequence x superscript t 2 1 d x t d t superscript t 2 1 sqrt 1 x 2 t quad x sqrt t 2 1 quad dx frac t dt sqrt t 2 1 yielding 1 x 2 x d x t 2 d t t 2 1 t 1 2 ln t 1 t 1 C t arcoth t C 1 superscript x 2 x d x superscript t 2 d t superscript t 2 1 t 1 2 t 1 t 1 C t arcoth t C int frac sqrt 1 x 2 x dx int frac t 2 dt t 2 1 t frac 1 2 ln frac t 1 t 1 C t arcoth t C see area functions and then 1 x 2 x d x 1 x 2 1 2 ln 1 x 2 1 1 x 2 1 C 1 x 2 ln x 1 1 x 2 C 1 superscript x 2 x d x 1 superscript x 2 1 2 1 superscript x 2 1 1 superscript x 2 1 C 1 superscript x 2 x 1 1 superscript x 2 C int frac sqrt 1 x 2 x dx sqrt 1 x 2 frac 1 2 ln frac sqrt 1 x 2 1 sqrt 1 x 2 1 C sqrt 1 x 2 ln frac x 1 sqrt 1 x 2 C Using this antiderivative one can obtain the arc length 1 For example if a 3 a 3 a sqrt 3 and b 15 b 15 b sqrt 15 the result is s 2 ln 3 5 s 2 3 5 s 2 ln frac 3 sqrt 5 As for finding the arc length of

    Original URL path: http://www.planetmath.org/arclengthoflogarithmiccurve (2016-04-25)
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  • arc length of parabola | planetmath.org
    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 3A26A09 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 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 arc length of parabola The parabola is one of the quite few plane curves the arc length of which is expressible in closed form other ones are line circle semicubical parabola logarithmic curve catenary tractrix cycloid clothoid astroid Nielsen s spiral logarithmic spiral Determining the arc length of ellipse and hyperbola leads to elliptic integrals We evaluate the length of the parabola y a x 2

    Original URL path: http://www.planetmath.org/arclengthofparabola (2016-04-25)
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  • Archimedes' cylinders in cube | planetmath.org
    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 3A26A09 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 3A26A09 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 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

    Original URL path: http://www.planetmath.org/archimedescylindersincube (2016-04-25)
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  • area bounded by arc and two lines | planetmath.org
    φ 2 d φ A 1 2 superscript subscript α β superscript r φ 2 d φ displaystyle A frac 1 2 int alpha beta r varphi 2 d varphi 1 Proof We fit between α α alpha and β β beta a set of values φ 1 φ 2 φ n 1 subscript φ 1 subscript φ 2 normal subscript φ n 1 displaystyle varphi 1 varphi 2 ldots varphi n 1 2 and denote α φ 0 α subscript φ 0 alpha varphi 0 β φ n β subscript φ n beta varphi n and think the line segments from the origin to each point of the curve corresponding the values φ i subscript φ i varphi i Then the region is divided into n n n parts For every part we form inscribed and circumscribed circular sector with the common tip in the origin and the radii along the lines φ φ i φ subscript φ i varphi varphi i The union of the inscribed sectors is contained in the region and the union of the circumscribed sectors contains the region The unions have the areas i 1 n 1 2 r i 2 φ i φ i 1 and i 1 n 1 2 R i 2 φ i φ i 1 superscript subscript i 1 n 1 2 superscript subscript r i 2 subscript φ i subscript φ i 1 and superscript subscript i 1 n 1 2 superscript subscript R i 2 subscript φ i subscript φ i 1 sum i 1 n frac 1 2 r i 2 varphi i varphi i 1 quad mbox and quad sum i 1 n frac 1 2 R i 2 varphi i varphi i 1 where r i subscript r i r i means the least and R i subscript R i R i the greatest value of r φ r φ r varphi on the interval φ i 1 φ i subscript φ i 1 subscript φ i varphi i 1 varphi i Hence the area A A A is between these sums for any division of the interval α β α β alpha beta with the values of 2 But by the definition of the Riemann integral we know that there is only one real number having this property for any division and that also the definite integral α β 1 2 r φ 2 d φ 1 2 α β r φ 2 d φ superscript subscript α β 1 2 superscript r φ 2 d φ 1 2 superscript subscript α β superscript r φ 2 d φ int alpha beta frac 1 2 r varphi 2 d varphi frac 1 2 int alpha beta r varphi 2 d varphi is between those sums Q E D Example 1 Determine the area A A A enclosed by the lemniscate of Bernoulli r cos 2 φ r 2 φ r sqrt cos 2 varphi psplot linecolor blue linewidth 0 02 1 4141 4144

    Original URL path: http://www.planetmath.org/areaboundedbyarcandtwolines (2016-04-25)
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  • area functions | planetmath.org
    hyperbolic cosine in Latin cosinus hyperbolicus is arcosh arcosh arcosh area cosini hyperbolici arcosh x ln x x 2 1 assign arcosh x x superscript x 2 1 arcosh x ln x sqrt x 2 1 It is defined for x 1 x 1 x geqq 1 The inverse function of the hyperbolic tangent in Latin tangens hyperbolica is artanh artanh artanh area tangentis hyperbolicae artanh x 1 2 ln 1 x 1 x assign artanh x 1 2 1 x 1 x artanh x frac 1 2 ln frac 1 x 1 x It is defined for 1 x 1 1 x 1 1 x 1 The inverse function of the hyperbolic cotangent in Latin cotangens hyperbolica is arcoth arcoth arcoth area cotangentis hyperbolicae arcoth x 1 2 ln x 1 x 1 assign arcoth x 1 2 x 1 x 1 arcoth x frac 1 2 ln frac x 1 x 1 It is defined for x 1 x 1 x 1 These four functions are denoted also by sinh 1 x superscript 1 x sinh 1 x cosh 1 x superscript 1 x cosh 1 x tanh 1 x superscript 1 x tanh 1 x and coth 1 x superscript hyperbolic cotangent 1 x coth 1 x Derivatives d d x arsinh x 1 x 2 1 d d x arsinh x 1 superscript x 2 1 frac d dx arsinh x frac 1 sqrt x 2 1 d d x arcosh x 1 x 2 1 d d x arcosh x 1 superscript x 2 1 frac d dx arcosh x frac 1 sqrt x 2 1 d d x artanh x 1 1 x 2 d d x artanh x 1 1 superscript x 2 frac d dx artanh x frac 1 1 x 2 d d x arcoth x 1 1 x 2 d d x arcoth x 1 1 superscript x 2 frac d dx arcoth x frac 1 1 x 2 The functions arsinh arsinh arsinh and artanh artanh artanh have the simple Taylor series arsinh x x 1 2 x 3 3 1 3 2 4 x 5 5 1 3 5 2 4 6 x 7 7 x 1 fragments arsinh x x 1 2 normal superscript x 3 3 normal 1 3 normal 2 4 normal superscript x 5 5 normal 1 3 5 normal 2 4 6 normal superscript x 7 7 normal fragments normal normal x normal 1 normal normal arsinh x x frac 1 2 cdot frac x 3 3 frac 1 cdot 3 2 cdot 4 cdot frac x 5 5 frac 1 cdot 3 cdot 5 2 cdot 4 cdot 6 cdot frac x 7 7 cdots quad x leqq 1 artanh x x x 3 3 x 5 5 x 7 7 x 1 fragments artanh x x superscript x 3 3 superscript x 5 5 superscript x 7 7 normal fragments normal normal x normal 1 normal normal artanh x x frac

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