archive-org.com » ORG » P » PLANETMATH.ORG

Total: 488

Choose link from "Titles, links and description words view":

Or switch to "Titles and links view".
  • first primitive Pythagorean triplets | planetmath.org
    2 superscript 96 2 superscript 265 2 247 2 96 2 265 2 231 2 160 2 281 2 superscript 231 2 superscript 160 2 superscript 281 2 231 2 160 2 281 2 207 2 224 2 305 2 superscript 207 2 superscript 224 2 superscript 305 2 207 2 224 2 305 2 175 2 288 2 337 2 superscript 175 2 superscript 288 2 superscript 337 2 175 2 288 2 337 2 135 2 352 2 377 2 superscript 135 2 superscript 352 2 superscript 377 2 135 2 352 2 377 2 87 2 416 2 425 2 superscript 87 2 superscript 416 2 superscript 425 2 87 2 416 2 425 2 31 2 480 2 481 2 superscript 31 2 superscript 480 2 superscript 481 2 31 2 480 2 481 2 285 2 68 2 293 2 superscript 285 2 superscript 68 2 superscript 293 2 285 2 68 2 293 2 273 2 136 2 305 2 superscript 273 2 superscript 136 2 superscript 305 2 273 2 136 2 305 2 253 2 204 2 325 2 superscript 253 2 superscript 204 2 superscript 325 2 253 2 204 2 325 2 225 2 272 2 353 2 superscript 225 2 superscript 272 2 superscript 353 2 225 2 272 2 353 2 189 2 340 2 389 2 superscript 189 2 superscript 340 2 superscript 389 2 189 2 340 2 389 2 145 2 408 2 433 2 superscript 145 2 superscript 408 2 superscript 433 2 145 2 408 2 433 2 93 2 476 2 485 2 superscript 93 2 superscript 476 2 superscript 485 2 93 2 476 2 485 2 33 2 544 2 545 2 superscript 33 2 superscript 544 2 superscript 545 2 33 2 544 2 545 2 323 2 36 2 325 2 superscript 323 2 superscript 36 2 superscript 325 2 323 2 36 2 325 2 299 2 180 2 349 2 superscript 299 2 superscript 180 2 superscript 349 2 299 2 180 2 349 2 275 2 252 2 373 2 superscript 275 2 superscript 252 2 superscript 373 2 275 2 252 2 373 2 203 2 396 2 445 2 superscript 203 2 superscript 396 2 superscript 445 2 203 2 396 2 445 2 155 2 468 2 493 2 superscript 155 2 superscript 468 2 superscript 493 2 155 2 468 2 493 2 35 2 612 2 613 2 superscript 35 2 superscript 612 2 superscript 613 2 35 2 612 2 613 2 357 2 76 2 365 2 superscript 357 2 superscript 76 2 superscript 365 2 357 2 76 2 365 2 345 2 152 2 377 2 superscript 345 2 superscript 152 2 superscript 377 2 345 2 152 2 377 2 325 2 228 2 397 2 superscript 325 2 superscript 228 2 superscript 397 2 325 2 228 2 397 2 297 2 304 2 425 2 superscript 297 2 superscript 304 2 superscript 425 2 297 2 304 2 425 2 261 2 380 2 461 2 superscript 261 2 superscript 380 2 superscript 461 2 261 2 380 2 461 2 217 2 456 2 505 2 superscript 217 2 superscript 456 2 superscript 505 2 217 2 456 2 505 2 165 2 532 2 557 2 superscript 165 2 superscript 532 2 superscript 557 2 165 2 532 2 557 2 105 2 608 2 617 2 superscript 105 2 superscript 608 2 superscript 617 2 105 2 608 2 617 2 37 2 684 2 685 2 superscript 37 2 superscript 684 2 superscript 685 2 37 2 684 2 685 2 399 2 40 2 401 2 superscript 399 2 superscript 40 2 superscript 401 2 399 2 40 2 401 2 391 2 120 2 409 2 superscript 391 2 superscript 120 2 superscript 409 2 391 2 120 2 409 2 351 2 280 2 449 2 superscript 351 2 superscript 280 2 superscript 449 2 351 2 280 2 449 2 319 2 360 2 481 2 superscript 319 2 superscript 360 2 superscript 481 2 319 2 360 2 481 2 279 2 440 2 521 2 superscript 279 2 superscript 440 2 superscript 521 2 279 2 440 2 521 2 231 2 520 2 569 2 superscript 231 2 superscript 520 2 superscript 569 2 231 2 520 2 569 2 111 2 680 2 689 2 superscript 111 2 superscript 680 2 superscript 689 2 111 2 680 2 689 2 39 2 760 2 761 2 superscript 39 2 superscript 760 2 superscript 761 2 39 2 760 2 761 2 437 2 84 2 445 2 superscript 437 2 superscript 84 2 superscript 445 2 437 2 84 2 445 2 425 2 168 2 457 2 superscript 425 2 superscript 168 2 superscript 457 2 425 2 168 2 457 2 377 2 336 2 505 2 superscript 377 2 superscript 336 2 superscript 505 2 377 2 336 2 505 2 341 2 420 2 541 2 superscript 341 2 superscript 420 2 superscript 541 2 341 2 420 2 541 2 185 2 672 2 697 2 superscript 185 2 superscript 672 2 superscript 697 2 185 2 672 2 697 2 41 2 840 2 841 2 superscript 41 2 superscript 840 2 superscript 841 2 41 2 840 2 841 2 483 2 44 2 485 2 superscript 483 2 superscript 44 2 superscript 485 2 483 2 44 2 485 2 475 2 132 2 493 2 superscript 475 2 superscript 132 2 superscript 493 2 475 2 132 2 493 2 459 2 220 2 509 2 superscript 459 2 superscript 220 2 superscript 509 2 459 2 220 2 509 2 435 2 308 2 533 2 superscript 435 2 superscript 308 2 superscript 533 2 435 2 308 2 533 2

    Original URL path: http://www.planetmath.org/firstprimitivepythagoreantriplets (2016-04-25)
    Open archived version from archive


  • flux of vector field | planetmath.org
    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 3A11 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 Primary tabs View active tab Coauthors PDF Source Edit flux of vector field Let U U x i U y j U z k normal U subscript U x normal i subscript U y normal j subscript U z normal k vec U U x vec i U y vec j U z vec k be a vector field in ℝ 3 superscript ℝ 3 mathbb R 3 and let a a a be a portion of some surface in the vector field Define one side of a a a to be positive if a a a is a closed surface then the positive side must be the outer surface of it For any surface element d a d a da of a a a the corresponding vectoral surface element is d a n d a d normal a normal n d a d vec a vec n da where n normal n vec n is the unit normal vector on the positive side of d a d a da The flux of the vector U normal U vec U through the surface a a a is the surface integral a U d a subscript a normal normal U d normal a int a vec U cdot d vec a Remark One can imagine that U normal U vec U represents the velocity vector of a flowing liquid suppose that the flow is stationary i e the velocity U normal U vec U depends only on the location not on the time Then the scalar product U d a normal normal U d normal a vec U cdot d vec a is the volume of the liquid flown per time unit through the surface element d a d a da it is positive or negative depending on whether the flow is from the negative side to the positive side or contrarily Example Let U x i 2 y j 3 z k normal U x normal i 2 y normal j 3 z normal k vec U x vec i 2y vec j 3z vec k and a a a be the portion of the plane x y x 1 x y x 1 x y x 1 in the first octant x 0 y 0 z 0 formulae sequence x 0 formulae sequence y 0 z 0 x geqq 0 y geqq 0 z geqq 0 with the positive normal away from the origin One has the constant unit normal vector n 1

    Original URL path: http://www.planetmath.org/fluxofvectorfield (2016-04-25)
    Open archived version from archive

  • formal congruence | planetmath.org
    subscript X 2 normal subscript X n f X 1 X 2 ldots X n and g X 1 X 2 X n g subscript X 1 subscript X 2 normal subscript X n g X 1 X 2 ldots X n with integer coefficients are said to be formally congruent modulo m m m denoted f X 1 X 2 X n g X 1 X 2 X n mod m annotated f subscript X 1 subscript X 2 normal subscript X n normal g subscript X 1 subscript X 2 normal subscript X n pmod m displaystyle f X 1 X 2 ldots X n underline equiv g X 1 X 2 ldots X n mathop rm mod m 1 iff all coefficients of the difference polynomial f g f g f g are divisible by m m m Remark 1 The formal congruence of polynomials is an equivalence relation in the set ℤ X 1 X 2 X n ℤ subscript X 1 subscript X 2 normal subscript X n mathbb Z X 1 X 2 ldots X n Remark 2 The formal congruence 1 implies that all integers substituted for the variables satisfy it in other words one can speak of an identical congruence However there are identical congruences that are not formal congruences e g X p X mod p superscript X p annotated X pmod p X p equiv X mathop rm mod p where p p p is a positive prime Examples 1 X Y p X p Y p mod p annotated superscript X Y p normal superscript X p superscript Y p pmod p X Y p underline equiv X p Y p mathop rm mod p when p p p is a positive prime number This result is easily proved with the binomial theorem cp the freshman s dream and may be by induction generalized to X 1 X 2 X n p X 1 p X 2 p X n p mod p annotated superscript subscript X 1 subscript X 2 normal subscript X n p normal superscript subscript X 1 p superscript subscript X 2 p normal superscript subscript X n p pmod p X 1 X 2 ldots X n p underline equiv X 1 p X 2 p ldots X n p mathop rm mod p If one substitutes in this formal congruence 1 for all X 1 subscript X 1 X 1 X 2 subscript X 2 X 2 X n subscript X n X n one obtains the congruence n p n mod p superscript n p annotated n pmod p n p equiv n mathop rm mod p similar substitution of 1 1 1 shows that the last congruence is valid also for negative integers n n n If it is supposed that p p p is not factor of n n n we have got the Fermat s little theorem 2 Let p p p be a positive prime number It is

    Original URL path: http://www.planetmath.org/formalcongruence (2016-04-25)
    Open archived version from archive

  • formal power series over field | planetmath.org
    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 3A11C08 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 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 Primary tabs View active tab Coauthors PDF Source Edit formal power series over field Theorem If K 𝐾 is a field then the ring K X 𝐾 delimited delimited 𝑋 of formal power series is a discrete valuation ring with X 𝑋 its unique maximal ideal Proof We show first that an arbitrary ideal I 𝐼 of K X 𝐾 delimited delimited 𝑋 is a principal ideal If I 0 𝐼 0 the thing is ready Therefore let I 0 𝐼 0 Take an element f X i 0 a i X i assign 𝑓 𝑋 superscript subscript 𝑖 0 subscript 𝑎 𝑖 superscript 𝑋 𝑖 of I 𝐼 such that it has the least possible amount of successive zero coefficients in its beginning let its first non zero coefficient be a k subscript 𝑎 𝑘 Then f X X k a k a k 1 X 𝑓 𝑋 superscript 𝑋 𝑘 subscript 𝑎 𝑘 subscript 𝑎 𝑘 1 𝑋 Here we have in the parentheses an invertible formal power series g X 𝑔 𝑋 whence get the equation X k f X g X 1 superscript 𝑋 𝑘 𝑓 𝑋 superscript delimited 𝑔 𝑋 1 implying X k I superscript 𝑋 𝑘 𝐼 and consequently X k I superscript 𝑋 𝑘 𝐼 For obtaining the reverse inclusion suppose that h X b n X n b n 1 X n 1 assign ℎ 𝑋 subscript 𝑏 𝑛 superscript 𝑋

    Original URL path: http://www.planetmath.org/formalpowerseriesoverfield (2016-04-25)
    Open archived version from archive

  • Fourier series in complex form and Fourier integral | planetmath.org
    a Riemann integrable real function f f f on the interval p p p p p p is f t a 0 2 n 1 a n cos n π t p b n sin n π t p f t subscript a 0 2 superscript subscript n 1 subscript a n n π t p subscript b n n π t p displaystyle f t frac a 0 2 sum n 1 infty left a n cos frac n pi t p b n sin frac n pi t p right 1 where the coefficients are a n 1 p p p f x cos n π t p d t b n 1 p p p f x sin n π t p d t formulae sequence subscript a n 1 p superscript subscript p p f x n π t p d t subscript b n 1 p superscript subscript p p f x n π t p d t displaystyle a n frac 1 p int p p f x cos frac n pi t p dt quad b n frac 1 p int p p f x sin frac n pi t p dt 2 If one expresses the cosines and sines via Euler formulas with exponential function the series 1 attains the form f t n c n e i n π t p f t superscript subscript n subscript c n superscript e i n π t p displaystyle f t sum n infty infty c n e frac in pi t p 3 The coefficients c n subscript c n c n could be obtained of a n subscript a n a n and b n subscript b n b n but they are comfortably derived directly by multiplying the equation 3 by e i m π t p superscript e i m π t p e frac im pi t p and integrating it from p p p to p p p One obtains c n 1 2 p p p f t e i n π t p d t n 0 1 2 fragments subscript c n 1 2 p superscript subscript p p f fragments normal t normal superscript e i n π t p d t italic fragments normal n 0 normal plus or minus 1 normal plus or minus 2 normal normal normal normal displaystyle c n frac 1 2p int p p f t e frac in pi t p dt qquad n 0 pm 1 pm 2 ldots 4 We may say that in 3 f t f t f t has been dissolved to sum of harmonics elementary waves c n e i n π t p subscript c n superscript e i n π t p c n e frac in pi t p with amplitudes c n subscript c n c n corresponding the frequencies n n n 0 2 Derivation of Fourier integral For seeing how the expansion 3 changes

    Original URL path: http://www.planetmath.org/fourierseriesincomplexformandfourierintegral (2016-04-25)
    Open archived version from archive

  • Fourier series of function of bounded variation | planetmath.org
    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 3A42A16 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 3A44A55 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 3A44A55 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 Fourier series of function of bounded variation If the real function f f f is of bounded variation on the interval π π π π pi pi then its Fourier series expansion a 0 2 n 1 a n cos n x b n sin n x subscript a 0 2 superscript subscript n 1 subscript a n n x subscript b n n x displaystyle frac a 0 2 sum n 1 infty a n cos nx b n sin nx 1 with the coefficients a n 1 π π π f x cos n x d x b n 1 π π π f x sin n x d x an 1π ππ fxcos nxdxbn 1π ππ fxsin nxdx displaystyle begin cases a n frac 1 pi int pi pi f x cos nx dx b n frac 1 pi int pi pi f x sin nx dx end cases 2 converges at every point of

    Original URL path: http://www.planetmath.org/fourierseriesoffunctionofboundedvariation (2016-04-25)
    Open archived version from archive

  • Fourier sine and cosine series | planetmath.org
    normal normal x equiv 2 left frac sin x 1 frac sin 2x 2 frac sin 3x 3 cdots right Remark 1 On the half interval 0 π 0 π 0 pi one can in any case expand each Riemann integrable function f f f both to a cosine series and to a sine series irrespective of how it is defined for the negative half interval or is it defined here at all Remark 2 On an interval p p p p p p instead of π π π π pi pi the Fourier coefficients of the series f x a 0 2 n 1 a n cos n π x p b n sin n π x p f x subscript a 0 2 superscript subscript n 1 subscript a n n π x p subscript b n n π x p f x frac a 0 2 sum n 1 infty left a n cos frac n pi x p b n sin frac n pi x p right have the expressions a n 2 p 0 p f x cos n π x p d x subscript a n 2 p superscript subscript 0 p f x n π x p d x displaystyle a n frac 2 p int 0 p f x cos frac n pi x p dx b n 0 subscript b n 0 b n 0 n for all n forall n if f f f is an even function b n 2 p 0 p f x sin n π x p d x subscript b n 2 p superscript subscript 0 p f x n π x p d x displaystyle b n frac 2 p int 0 p f x sin frac n pi x p dx a n 0 subscript a n 0 a n 0 n for all n forall n if f f f is an odd function Example Expand the identity function x x maps to x x x mapsto x to a Fourier cosine series on 0 π 0 π 0 pi This odd function may be replaced with the even function f x x normal f maps to x x f x mapsto x Then we get a 0 2 π 0 π x d x π subscript a 0 2 π superscript subscript 0 π x d x π a 0 frac 2 pi int 0 pi x dx pi and integrating by parts a n 2 π 0 π x cos n x d x 2 π 0 π x sin n x n 0 π sin n x n d x 2 π 0 π cos n x n 2 2 π n 2 1 n 1 fragments subscript a n 2 π superscript subscript 0 π x n x d x 2 π fragments normal superscript subscript normal 0 π x n x n superscript subscript 0 π n x n d x normal 2 π superscript subscript normal

    Original URL path: http://www.planetmath.org/fouriersineandcosineseries (2016-04-25)
    Open archived version from archive

  • fraction power | planetmath.org
    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 Let m m m be an integer and n n n a positive factor of m m m If x x x is a positive real number we may write the identical equation x m n n x m n n x m superscript superscript x m n n superscript x normal m n n superscript x m x frac m n n x frac m n cdot n x m and therefore the definition of n th superscript n th n mathrm th root gives the formula x m n x m n n superscript x m superscript x m n displaystyle sqrt n x m x frac m n 1 Here the exponent m n m n frac m n is an integer For enabling the validity of 1 for the cases where n n n does not divide m m m we must set the following Definition Let m n m n frac m n be a fractional number i e an integer m m m not divisible by the integer n n n which latter we assume to be positive For any positive real number x x x we define the fraction power x m n superscript x m n x frac m n as the n th superscript n th n mathrm th root x m n x m n assign superscript x m n n superscript x m displaystyle x frac m n sqrt n x

    Original URL path: http://www.planetmath.org/fractionpower (2016-04-25)
    Open archived version from archive



  •