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Simplificación de expresiones/ Descomposición en fracciones simples/ sqrt(4*(-9*cos(4*pi*K/3)/(pi^2*K^2)+3*sin(2*pi*K/3)/(pi*K)-9*cos(2*pi*K/3)/(2*pi^2*K^2)-3*sin(4*pi*K/3)/(2*pi*K)+27*sin(2*pi*K/3)/(4*pi^3*K^3)+27*sin(4*pi*K/3)/(4*pi^3*K^3))^2/9+4*(-3*cos(2*pi*K/3)/(pi*K)+9*sin(4*pi*K/3)/(pi^2*K^2)-27*cos(2*pi*K/3)/(4*pi^3*K^3)-9*sin(2*pi*K/3)/(2*pi^2*K^2)-3*cos(4*pi*K/3)/(2*pi*K)+27*cos(4*pi*K/3)/(4*pi^3*K^3))^2/9)

¿Cómo vas a descomponer esta sqrt(4*(-9*cos(4*pi*K/3)/(pi^2*K^2)+3*sin(2*pi*K/3)/(pi*K)-9*cos(2*pi*K/3)/(2*pi^2*K^2)-3*sin(4*pi*K/3)/(2*pi*K)+27*sin(2*pi*K/3)/(4*pi^3*K^3)+27*sin(4*pi*K/3)/(4*pi^3*K^3))^2/9+4*(-3*cos(2*pi*K/3)/(pi*K)+9*sin(4*pi*K/3)/(pi^2*K^2)-27*cos(2*pi*K/3)/(4*pi^3*K^3)-9*sin(2*pi*K/3)/(2*pi^2*K^2)-3*cos(4*pi*K/3)/(2*pi*K)+27*cos(4*pi*K/3)/(4*pi^3*K^3))^2/9) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
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         /                                                                                                     2                                                                                                       2 
        /    /      /4*pi*k\        /2*pi*k\        /2*pi*k\        /4*pi*k\         /2*pi*k\         /4*pi*k\\      /      /2*pi*k\        /4*pi*k\         /2*pi*k\        /2*pi*k\        /4*pi*k\         /4*pi*k\\  
       /     |-9*cos|------|   3*sin|------|   9*cos|------|   3*sin|------|   27*sin|------|   27*sin|------||      |-3*cos|------|   9*sin|------|   27*cos|------|   9*sin|------|   3*cos|------|   27*cos|------||  
      /      |      \  3   /        \  3   /        \  3   /        \  3   /         \  3   /         \  3   /|      |      \  3   /        \  3   /         \  3   /        \  3   /        \  3   /         \  3   /|  
     /     4*|-------------- + ------------- - ------------- - ------------- + -------------- + --------------|    4*|-------------- + ------------- - -------------- - ------------- - ------------- + --------------|  
    /        |      2  2            pi*k              2  2         2*pi*k             3  3             3  3   |      |     pi*k              2  2             3  3             2  2         2*pi*k             3  3   |  
   /         \    pi *k                           2*pi *k                         4*pi *k          4*pi *k    /      \                     pi *k          4*pi *k          2*pi *k                         4*pi *k    /  
  /        ----------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------- 
\/                                                           9                                                                                                       9
4(((((9sin⁡(4πk3)π2k2+(−1)3cos⁡(2πk3)πk)−27cos⁡(2πk3)4π3k3)−9sin⁡(2πk3)2π2k2)−3cos⁡(4πk3)2πk)+27cos⁡(4πk3)4π3k3)29+4((((((−1)9cos⁡(4πk3)π2k2+3sin⁡(2πk3)πk)−9cos⁡(2πk3)2π2k2)−3sin⁡(4πk3)2πk)+27sin⁡(2πk3)4π3k3)+27sin⁡(4πk3)4π3k3)29\sqrt{\frac{4 \left(\left(\left(\left(\left(\frac{9 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{\left(-1\right) 3 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k}\right) - \frac{27 \cos{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right) - \frac{9 \sin{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}}\right) - \frac{3 \cos{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k}\right) + \frac{27 \cos{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9} + \frac{4 \left(\left(\left(\left(\left(\frac{\left(-1\right) 9 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{3 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi k}\right) - \frac{9 \cos{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}}\right) - \frac{3 \sin{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k}\right) + \frac{27 \sin{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right) + \frac{27 \sin{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9}} 
sqrt((4*((-9*cos(((4*pi)*k)/3))/((pi^2*k^2)) + (3*sin(((2*pi)*k)/3))/((pi*k)) - 9*cos(((2*pi)*k)/3)/((2*pi^2)*k^2) - 3*sin(((4*pi)*k)/3)/((2*pi)*k) + (27*sin(((2*pi)*k)/3))/(((4*pi^3)*k^3)) + (27*sin(((4*pi)*k)/3))/(((4*pi^3)*k^3)))^2)/9 + (4*((-3*cos(((2*pi)*k)/3))/((pi*k)) + (9*sin(((4*pi)*k)/3))/((pi^2*k^2)) - 27*cos(((2*pi)*k)/3)/((4*pi^3)*k^3) - 9*sin(((2*pi)*k)/3)/((2*pi^2)*k^2) - 3*cos(((4*pi)*k)/3)/((2*pi)*k) + (27*cos(((4*pi)*k)/3))/(((4*pi^3)*k^3)))^2)/9)
Descomposición de una fracción [src] 
sqrt(cos(4*pi*k/3)^2/(pi^2*k^2) + sin(4*pi*k/3)^2/(pi^2*k^2) + 4*cos(2*pi*k/3)^2/(pi^2*k^2) + 4*sin(2*pi*k/3)^2/(pi^2*k^2) + 27*cos(2*pi*k/3)^2/(pi^4*k^4) + 27*cos(4*pi*k/3)^2/(pi^4*k^4) + 27*sin(2*pi*k/3)^2/(pi^4*k^4) + 27*sin(4*pi*k/3)^2/(pi^4*k^4) + 81*cos(2*pi*k/3)^2/(4*pi^6*k^6) + 81*cos(4*pi*k/3)^2/(4*pi^6*k^6) + 81*sin(2*pi*k/3)^2/(4*pi^6*k^6) + 81*sin(4*pi*k/3)^2/(4*pi^6*k^6) - 81*cos(2*pi*k/3)*sin(4*pi*k/3)/(pi^5*k^5) - 81*cos(4*pi*k/3)*sin(2*pi*k/3)/(pi^5*k^5) - 27*sin(2*pi*k/3)*sin(4*pi*k/3)/(pi^4*k^4) - 18*cos(2*pi*k/3)*sin(4*pi*k/3)/(pi^3*k^3) - 18*cos(4*pi*k/3)*sin(2*pi*k/3)/(pi^3*k^3) - 4*sin(2*pi*k/3)*sin(4*pi*k/3)/(pi^2*k^2) + 4*cos(2*pi*k/3)*cos(4*pi*k/3)/(pi^2*k^2) + 27*cos(2*pi*k/3)*cos(4*pi*k/3)/(pi^4*k^4) - 81*cos(2*pi*k/3)*cos(4*pi*k/3)/(2*pi^6*k^6) + 81*sin(2*pi*k/3)*sin(4*pi*k/3)/(2*pi^6*k^6))
4sin⁡2(2πk3)π2k2−4sin⁡(2πk3)sin⁡(4πk3)π2k2+sin⁡2(4πk3)π2k2+4cos⁡2(2πk3)π2k2+4cos⁡(2πk3)cos⁡(4πk3)π2k2+cos⁡2(4πk3)π2k2−18sin⁡(2πk3)cos⁡(4πk3)π3k3−18sin⁡(4πk3)cos⁡(2πk3)π3k3+27sin⁡2(2πk3)π4k4−27sin⁡(2πk3)sin⁡(4πk3)π4k4+27sin⁡2(4πk3)π4k4+27cos⁡2(2πk3)π4k4+27cos⁡(2πk3)cos⁡(4πk3)π4k4+27cos⁡2(4πk3)π4k4−81sin⁡(2πk3)cos⁡(4πk3)π5k5−81sin⁡(4πk3)cos⁡(2πk3)π5k5+81sin⁡2(2πk3)4π6k6+81sin⁡(2πk3)sin⁡(4πk3)2π6k6+81sin⁡2(4πk3)4π6k6+81cos⁡2(2πk3)4π6k6−81cos⁡(2πk3)cos⁡(4πk3)2π6k6+81cos⁡2(4πk3)4π6k6\sqrt{\frac{4 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{4 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{\sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{4 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{4 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{\cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{18 \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{3} k^{3}} - \frac{18 \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{3} k^{3}} + \frac{27 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{4} k^{4}} - \frac{27 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} - \frac{81 \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{5} k^{5}} - \frac{81 \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{5} k^{5}} + \frac{81 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{6} k^{6}} + \frac{81 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{2 \pi^{6} k^{6}} + \frac{81 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{6} k^{6}} + \frac{81 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{6} k^{6}} - \frac{81 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{2 \pi^{6} k^{6}} + \frac{81 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{6} k^{6}}} 
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      /    2/4*pi*k\      2/4*pi*k\        2/2*pi*k\        2/2*pi*k\         2/2*pi*k\         2/4*pi*k\         2/2*pi*k\         2/4*pi*k\         2/2*pi*k\         2/4*pi*k\         2/2*pi*k\         2/4*pi*k\         /2*pi*k\    /4*pi*k\         /4*pi*k\    /2*pi*k\         /2*pi*k\    /4*pi*k\         /2*pi*k\    /4*pi*k\         /4*pi*k\    /2*pi*k\        /2*pi*k\    /4*pi*k\        /2*pi*k\    /4*pi*k\         /2*pi*k\    /4*pi*k\         /2*pi*k\    /4*pi*k\         /2*pi*k\    /4*pi*k\ 
     /  cos |------|   sin |------|   4*cos |------|   4*sin |------|   27*cos |------|   27*cos |------|   27*sin |------|   27*sin |------|   81*cos |------|   81*cos |------|   81*sin |------|   81*sin |------|   81*cos|------|*sin|------|   81*cos|------|*sin|------|   27*sin|------|*sin|------|   18*cos|------|*sin|------|   18*cos|------|*sin|------|   4*sin|------|*sin|------|   4*cos|------|*cos|------|   27*cos|------|*cos|------|   81*cos|------|*cos|------|   81*sin|------|*sin|------| 
    /       \  3   /       \  3   /         \  3   /         \  3   /          \  3   /          \  3   /          \  3   /          \  3   /          \  3   /          \  3   /          \  3   /          \  3   /         \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   /        \  3   /    \  3   /        \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   / 
   /    ------------ + ------------ + -------------- + -------------- + --------------- + --------------- + --------------- + --------------- + --------------- + --------------- + --------------- + --------------- - -------------------------- - -------------------------- - -------------------------- - -------------------------- - -------------------------- - ------------------------- + ------------------------- + -------------------------- - -------------------------- + -------------------------- 
  /          2  2           2  2            2  2             2  2              4  4              4  4              4  4              4  4               6  6              6  6              6  6              6  6                  5  5                         5  5                         4  4                         3  3                         3  3                         2  2                        2  2                        4  4                          6  6                         6  6          
\/         pi *k          pi *k           pi *k            pi *k             pi *k             pi *k             pi *k             pi *k            4*pi *k           4*pi *k           4*pi *k           4*pi *k                 pi *k                        pi *k                        pi *k                        pi *k                        pi *k                        pi *k                       pi *k                       pi *k                       2*pi *k                      2*pi *k
Simplificación general [src] 
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       /                                                                                                                    2                                                                                                          2 
      /  /       /4*pi*k\        /2*pi*k\       2  2 /     /2*pi*k\      /4*pi*k\\          /       /4*pi*k\      /2*pi*k\\\    /     /2*pi*k\        /4*pi*k\          /     /4*pi*k\      /2*pi*k\\        2  2    3/pi*k\    /pi*k\\  
     /   |- 9*cos|------| + 9*cos|------| + 2*pi *k *|2*cos|------| + cos|------|| + 6*pi*k*|- 2*sin|------| + sin|------|||  + |9*sin|------| + 9*sin|------| - 6*pi*k*|2*cos|------| + cos|------|| + 16*pi *k *sin |----|*cos|----||  
    /    \       \  3   /        \  3   /            \     \  3   /      \  3   //          \       \  3   /      \  3   ///    \     \  3   /        \  3   /          \     \  3   /      \  3   //                 \ 3  /    \ 3  //  
   /     ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 
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                                                                                                                      3                                                                                                                  
                                                                                                                  2*pi
(2π2k2(2cos⁡(2πk3)+cos⁡(4πk3))+6πk(sin⁡(2πk3)−2sin⁡(4πk3))+9cos⁡(2πk3)−9cos⁡(4πk3))2+(16π2k2sin⁡3(πk3)cos⁡(πk3)−6πk(cos⁡(2πk3)+2cos⁡(4πk3))+9sin⁡(2πk3)+9sin⁡(4πk3))2k62π3\frac{\sqrt{\frac{\left(2 \pi^{2} k^{2} \left(2 \cos{\left(\frac{2 \pi k}{3} \right)} + \cos{\left(\frac{4 \pi k}{3} \right)}\right) + 6 \pi k \left(\sin{\left(\frac{2 \pi k}{3} \right)} - 2 \sin{\left(\frac{4 \pi k}{3} \right)}\right) + 9 \cos{\left(\frac{2 \pi k}{3} \right)} - 9 \cos{\left(\frac{4 \pi k}{3} \right)}\right)^{2} + \left(16 \pi^{2} k^{2} \sin^{3}{\left(\frac{\pi k}{3} \right)} \cos{\left(\frac{\pi k}{3} \right)} - 6 \pi k \left(\cos{\left(\frac{2 \pi k}{3} \right)} + 2 \cos{\left(\frac{4 \pi k}{3} \right)}\right) + 9 \sin{\left(\frac{2 \pi k}{3} \right)} + 9 \sin{\left(\frac{4 \pi k}{3} \right)}\right)^{2}}{k^{6}}}}{2 \pi^{3}} 
sqrt(((-9*cos(4*pi*k/3) + 9*cos(2*pi*k/3) + 2*pi^2*k^2*(2*cos(2*pi*k/3) + cos(4*pi*k/3)) + 6*pi*k*(-2*sin(4*pi*k/3) + sin(2*pi*k/3)))^2 + (9*sin(2*pi*k/3) + 9*sin(4*pi*k/3) - 6*pi*k*(2*cos(4*pi*k/3) + cos(2*pi*k/3)) + 16*pi^2*k^2*sin(pi*k/3)^3*cos(pi*k/3))^2)/k^6)/(2*pi^3)
Respuesta numérica [src] 
0.636619772367581*((sin(((2*pi)*k)/3)/k + 0.22797266319526*sin(((2*pi)*k)/3)/k^3 + 0.22797266319526*sin(((4*pi)*k)/3)/k^3 - 0.5*sin(((4*pi)*k)/3)/k - 0.477464829275686*cos(((2*pi)*k)/3)/k^2 - 0.954929658551372*cos(((4*pi)*k)/3)/k^2)^2 + (-cos(((2*pi)*k)/3)/k + 0.954929658551372*sin(((4*pi)*k)/3)/k^2 + 0.22797266319526*cos(((4*pi)*k)/3)/k^3 - 0.5*cos(((4*pi)*k)/3)/k - 0.477464829275686*sin(((2*pi)*k)/3)/k^2 - 0.22797266319526*cos(((2*pi)*k)/3)/k^3)^2)^0.5
0.636619772367581*((sin(((2*pi)*k)/3)/k + 0.22797266319526*sin(((2*pi)*k)/3)/k^3 + 0.22797266319526*sin(((4*pi)*k)/3)/k^3 - 0.5*sin(((4*pi)*k)/3)/k - 0.477464829275686*cos(((2*pi)*k)/3)/k^2 - 0.954929658551372*cos(((4*pi)*k)/3)/k^2)^2 + (-cos(((2*pi)*k)/3)/k + 0.954929658551372*sin(((4*pi)*k)/3)/k^2 + 0.22797266319526*cos(((4*pi)*k)/3)/k^3 - 0.5*cos(((4*pi)*k)/3)/k - 0.477464829275686*sin(((2*pi)*k)/3)/k^2 - 0.22797266319526*cos(((2*pi)*k)/3)/k^3)^2)^0.5
Denominador común [src] 
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      /      2/4*pi*k\        2/4*pi*k\         2/2*pi*k\         2/2*pi*k\         /2*pi*k\    /4*pi*k\         /2*pi*k\    /4*pi*k\         2/2*pi*k\         2/4*pi*k\         2/2*pi*k\         2/4*pi*k\          2/2*pi*k\          2/4*pi*k\          2/2*pi*k\          2/4*pi*k\          /2*pi*k\    /4*pi*k\          /4*pi*k\    /2*pi*k\          /2*pi*k\    /4*pi*k\          /2*pi*k\    /4*pi*k\         /2*pi*k\    /4*pi*k\         /4*pi*k\    /2*pi*k\          /2*pi*k\    /4*pi*k\          /2*pi*k\    /4*pi*k\ 
     /  4*cos |------|   4*sin |------|   16*cos |------|   16*sin |------|   16*sin|------|*sin|------|   16*cos|------|*cos|------|   81*cos |------|   81*cos |------|   81*sin |------|   81*sin |------|   108*cos |------|   108*cos |------|   108*sin |------|   108*sin |------|   324*cos|------|*sin|------|   324*cos|------|*sin|------|   162*cos|------|*cos|------|   108*sin|------|*sin|------|   72*cos|------|*sin|------|   72*cos|------|*sin|------|   108*cos|------|*cos|------|   162*sin|------|*sin|------| 
    /         \  3   /         \  3   /          \  3   /          \  3   /         \  3   /    \  3   /         \  3   /    \  3   /          \  3   /          \  3   /          \  3   /          \  3   /           \  3   /           \  3   /           \  3   /           \  3   /          \  3   /    \  3   /          \  3   /    \  3   /          \  3   /    \  3   /          \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   /          \  3   /    \  3   /          \  3   /    \  3   / 
   /    -------------- + -------------- + --------------- + --------------- - -------------------------- + -------------------------- + --------------- + --------------- + --------------- + --------------- + ---------------- + ---------------- + ---------------- + ---------------- - --------------------------- - --------------------------- - --------------------------- - --------------------------- - -------------------------- - -------------------------- + --------------------------- + --------------------------- 
  /            2                2                 2                 2                      2                            2                      4  6              4  6              4  6              4  6              2  4               2  4               2  4               2  4                     3  5                          3  5                          4  6                          2  4                           3                            3                           2  4                          4  6           
\/            k                k                 k                 k                      k                            k                     pi *k             pi *k             pi *k             pi *k             pi *k              pi *k              pi *k              pi *k                    pi *k                         pi *k                         pi *k                         pi *k                        pi*k                         pi*k                          pi *k                         pi *k            
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                                                                                                                                                                                                                                                                  2*pi
16sin⁡2(2πk3)k2−16sin⁡(2πk3)sin⁡(4πk3)k2+4sin⁡2(4πk3)k2+16cos⁡2(2πk3)k2+16cos⁡(2πk3)cos⁡(4πk3)k2+4cos⁡2(4πk3)k2−72sin⁡(2πk3)cos⁡(4πk3)πk3−72sin⁡(4πk3)cos⁡(2πk3)πk3+108sin⁡2(2πk3)π2k4−108sin⁡(2πk3)sin⁡(4πk3)π2k4+108sin⁡2(4πk3)π2k4+108cos⁡2(2πk3)π2k4+108cos⁡(2πk3)cos⁡(4πk3)π2k4+108cos⁡2(4πk3)π2k4−324sin⁡(2πk3)cos⁡(4πk3)π3k5−324sin⁡(4πk3)cos⁡(2πk3)π3k5+81sin⁡2(2πk3)π4k6+162sin⁡(2πk3)sin⁡(4πk3)π4k6+81sin⁡2(4πk3)π4k6+81cos⁡2(2πk3)π4k6−162cos⁡(2πk3)cos⁡(4πk3)π4k6+81cos⁡2(4πk3)π4k62π\frac{\sqrt{\frac{16 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{2}} - \frac{16 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{4 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{16 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{2}} + \frac{16 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{4 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} - \frac{72 \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi k^{3}} - \frac{72 \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k^{3}} + \frac{108 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{4}} - \frac{108 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{4}} + \frac{108 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{4}} + \frac{108 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{4}} + \frac{108 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{4}} + \frac{108 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{4}} - \frac{324 \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{3} k^{5}} - \frac{324 \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{3} k^{5}} + \frac{81 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{4} k^{6}} + \frac{162 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{6}} + \frac{81 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{6}} + \frac{81 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{4} k^{6}} - \frac{162 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{6}} + \frac{81 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{6}}}}{2 \pi} 
sqrt(4*cos(4*pi*k/3)^2/k^2 + 4*sin(4*pi*k/3)^2/k^2 + 16*cos(2*pi*k/3)^2/k^2 + 16*sin(2*pi*k/3)^2/k^2 - 16*sin(2*pi*k/3)*sin(4*pi*k/3)/k^2 + 16*cos(2*pi*k/3)*cos(4*pi*k/3)/k^2 + 81*cos(2*pi*k/3)^2/(pi^4*k^6) + 81*cos(4*pi*k/3)^2/(pi^4*k^6) + 81*sin(2*pi*k/3)^2/(pi^4*k^6) + 81*sin(4*pi*k/3)^2/(pi^4*k^6) + 108*cos(2*pi*k/3)^2/(pi^2*k^4) + 108*cos(4*pi*k/3)^2/(pi^2*k^4) + 108*sin(2*pi*k/3)^2/(pi^2*k^4) + 108*sin(4*pi*k/3)^2/(pi^2*k^4) - 324*cos(2*pi*k/3)*sin(4*pi*k/3)/(pi^3*k^5) - 324*cos(4*pi*k/3)*sin(2*pi*k/3)/(pi^3*k^5) - 162*cos(2*pi*k/3)*cos(4*pi*k/3)/(pi^4*k^6) - 108*sin(2*pi*k/3)*sin(4*pi*k/3)/(pi^2*k^4) - 72*cos(2*pi*k/3)*sin(4*pi*k/3)/(pi*k^3) - 72*cos(4*pi*k/3)*sin(2*pi*k/3)/(pi*k^3) + 108*cos(2*pi*k/3)*cos(4*pi*k/3)/(pi^2*k^4) + 162*sin(2*pi*k/3)*sin(4*pi*k/3)/(pi^4*k^6))/(2*pi)
Unión de expresiones racionales [src] 
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       /                                                                                                                   2                                                                                                                            2 
      /  /     /2*pi*k\        /4*pi*k\          /       /4*pi*k\        /2*pi*k\           /4*pi*k\             /2*pi*k\\\    /       /2*pi*k\        /4*pi*k\             /2*pi*k\       2  2    /4*pi*k\          /     /4*pi*k\           /2*pi*k\\\  
     /   |9*sin|------| + 9*sin|------| + 2*pi*k*|- 6*cos|------| - 3*cos|------| - pi*k*sin|------| + 2*pi*k*sin|------|||  + |- 9*cos|------| + 9*cos|------| - 6*pi*k*sin|------| - 2*pi *k *cos|------| + 4*pi*k*|3*sin|------| - pi*k*cos|------|||  
    /    \     \  3   /        \  3   /          \       \  3   /        \  3   /           \  3   /             \  3   ///    \       \  3   /        \  3   /             \  3   /               \  3   /          \     \  3   /           \  3   ///  
   /     ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ 
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                                                                                                                              3                                                                                                                           
                                                                                                                          2*pi
(2πk(2πksin⁡(2πk3)−πksin⁡(4πk3)−3cos⁡(2πk3)−6cos⁡(4πk3))+9sin⁡(2πk3)+9sin⁡(4πk3))2+(−2π2k2cos⁡(4πk3)+4πk(−πkcos⁡(2πk3)+3sin⁡(4πk3))−6πksin⁡(2πk3)−9cos⁡(2πk3)+9cos⁡(4πk3))2k62π3\frac{\sqrt{\frac{\left(2 \pi k \left(2 \pi k \sin{\left(\frac{2 \pi k}{3} \right)} - \pi k \sin{\left(\frac{4 \pi k}{3} \right)} - 3 \cos{\left(\frac{2 \pi k}{3} \right)} - 6 \cos{\left(\frac{4 \pi k}{3} \right)}\right) + 9 \sin{\left(\frac{2 \pi k}{3} \right)} + 9 \sin{\left(\frac{4 \pi k}{3} \right)}\right)^{2} + \left(- 2 \pi^{2} k^{2} \cos{\left(\frac{4 \pi k}{3} \right)} + 4 \pi k \left(- \pi k \cos{\left(\frac{2 \pi k}{3} \right)} + 3 \sin{\left(\frac{4 \pi k}{3} \right)}\right) - 6 \pi k \sin{\left(\frac{2 \pi k}{3} \right)} - 9 \cos{\left(\frac{2 \pi k}{3} \right)} + 9 \cos{\left(\frac{4 \pi k}{3} \right)}\right)^{2}}{k^{6}}}}{2 \pi^{3}} 
sqrt(((9*sin(2*pi*k/3) + 9*sin(4*pi*k/3) + 2*pi*k*(-6*cos(4*pi*k/3) - 3*cos(2*pi*k/3) - pi*k*sin(4*pi*k/3) + 2*pi*k*sin(2*pi*k/3)))^2 + (-9*cos(2*pi*k/3) + 9*cos(4*pi*k/3) - 6*pi*k*sin(2*pi*k/3) - 2*pi^2*k^2*cos(4*pi*k/3) + 4*pi*k*(3*sin(4*pi*k/3) - pi*k*cos(2*pi*k/3)))^2)/k^6)/(2*pi^3)
Combinatoria [src] 
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         /                                                                                             2/2*pi*k\         2/4*pi*k\         2/2*pi*k\         2/4*pi*k\          /2*pi*k\    /4*pi*k\                                                                           2    2/2*pi*k\         2    2/4*pi*k\         2    2/2*pi*k\         2    2/4*pi*k\          /2*pi*k\    /4*pi*k\             /2*pi*k\    /4*pi*k\             /4*pi*k\    /2*pi*k\         2    /2*pi*k\    /4*pi*k\        3    /2*pi*k\    /4*pi*k\        3    /4*pi*k\    /2*pi*k\         2    /2*pi*k\    /4*pi*k\ 
        /                                                                                        81*cos |------|   81*cos |------|   81*sin |------|   81*sin |------|   162*cos|------|*cos|------|                                                                     108*pi *cos |------|   108*pi *cos |------|   108*pi *sin |------|   108*pi *sin |------|   162*sin|------|*sin|------|   324*pi*cos|------|*sin|------|   324*pi*cos|------|*sin|------|   108*pi *sin|------|*sin|------|   72*pi *cos|------|*sin|------|   72*pi *cos|------|*sin|------|   108*pi *cos|------|*cos|------| 
       /       4    2/4*pi*k\       4    2/4*pi*k\        4    2/2*pi*k\        4    2/2*pi*k\          \  3   /          \  3   /          \  3   /          \  3   /          \  3   /    \  3   /        4    /2*pi*k\    /4*pi*k\        4    /2*pi*k\    /4*pi*k\               \  3   /               \  3   /               \  3   /               \  3   /          \  3   /    \  3   /             \  3   /    \  3   /             \  3   /    \  3   /              \  3   /    \  3   /             \  3   /    \  3   /             \  3   /    \  3   /              \  3   /    \  3   / 
      /    4*pi *cos |------| + 4*pi *sin |------| + 16*pi *cos |------| + 16*pi *sin |------| + --------------- + --------------- + --------------- + --------------- - --------------------------- - 16*pi *sin|------|*sin|------| + 16*pi *cos|------|*cos|------| + -------------------- + -------------------- + -------------------- + -------------------- + --------------------------- - ------------------------------ - ------------------------------ - ------------------------------- - ------------------------------ - ------------------------------ + ------------------------------- 
     /               \  3   /             \  3   /              \  3   /              \  3   /           4                 4                 4                 4                       4                         \  3   /    \  3   /             \  3   /    \  3   /             2                      2                      2                      2                          4                              3                                3                                 2                               k                                k                                  2               
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                                                                                                                                                                                                                                                                                                      3                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                                                                  2*pi
16π4sin⁡2(2πk3)−16π4sin⁡(2πk3)sin⁡(4πk3)+4π4sin⁡2(4πk3)+16π4cos⁡2(2πk3)+16π4cos⁡(2πk3)cos⁡(4πk3)+4π4cos⁡2(4πk3)−72π3sin⁡(2πk3)cos⁡(4πk3)k−72π3sin⁡(4πk3)cos⁡(2πk3)k+108π2sin⁡2(2πk3)k2−108π2sin⁡(2πk3)sin⁡(4πk3)k2+108π2sin⁡2(4πk3)k2+108π2cos⁡2(2πk3)k2+108π2cos⁡(2πk3)cos⁡(4πk3)k2+108π2cos⁡2(4πk3)k2−324πsin⁡(2πk3)cos⁡(4πk3)k3−324πsin⁡(4πk3)cos⁡(2πk3)k3+81sin⁡2(2πk3)k4+162sin⁡(2πk3)sin⁡(4πk3)k4+81sin⁡2(4πk3)k4+81cos⁡2(2πk3)k4−162cos⁡(2πk3)cos⁡(4πk3)k4+81cos⁡2(4πk3)k4k22π3\frac{\sqrt{\frac{16 \pi^{4} \sin^{2}{\left(\frac{2 \pi k}{3} \right)} - 16 \pi^{4} \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)} + 4 \pi^{4} \sin^{2}{\left(\frac{4 \pi k}{3} \right)} + 16 \pi^{4} \cos^{2}{\left(\frac{2 \pi k}{3} \right)} + 16 \pi^{4} \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)} + 4 \pi^{4} \cos^{2}{\left(\frac{4 \pi k}{3} \right)} - \frac{72 \pi^{3} \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{k} - \frac{72 \pi^{3} \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{k} + \frac{108 \pi^{2} \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{2}} - \frac{108 \pi^{2} \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{108 \pi^{2} \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{108 \pi^{2} \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{2}} + \frac{108 \pi^{2} \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{108 \pi^{2} \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} - \frac{324 \pi \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{k^{3}} - \frac{324 \pi \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{k^{3}} + \frac{81 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{4}} + \frac{162 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{k^{4}} + \frac{81 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{4}} + \frac{81 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{4}} - \frac{162 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{k^{4}} + \frac{81 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{4}}}{k^{2}}}}{2 \pi^{3}} 
sqrt((4*pi^4*cos(4*pi*k/3)^2 + 4*pi^4*sin(4*pi*k/3)^2 + 16*pi^4*cos(2*pi*k/3)^2 + 16*pi^4*sin(2*pi*k/3)^2 + 81*cos(2*pi*k/3)^2/k^4 + 81*cos(4*pi*k/3)^2/k^4 + 81*sin(2*pi*k/3)^2/k^4 + 81*sin(4*pi*k/3)^2/k^4 - 162*cos(2*pi*k/3)*cos(4*pi*k/3)/k^4 - 16*pi^4*sin(2*pi*k/3)*sin(4*pi*k/3) + 16*pi^4*cos(2*pi*k/3)*cos(4*pi*k/3) + 108*pi^2*cos(2*pi*k/3)^2/k^2 + 108*pi^2*cos(4*pi*k/3)^2/k^2 + 108*pi^2*sin(2*pi*k/3)^2/k^2 + 108*pi^2*sin(4*pi*k/3)^2/k^2 + 162*sin(2*pi*k/3)*sin(4*pi*k/3)/k^4 - 324*pi*cos(2*pi*k/3)*sin(4*pi*k/3)/k^3 - 324*pi*cos(4*pi*k/3)*sin(2*pi*k/3)/k^3 - 108*pi^2*sin(2*pi*k/3)*sin(4*pi*k/3)/k^2 - 72*pi^3*cos(2*pi*k/3)*sin(4*pi*k/3)/k - 72*pi^3*cos(4*pi*k/3)*sin(2*pi*k/3)/k + 108*pi^2*cos(2*pi*k/3)*cos(4*pi*k/3)/k^2)/k^2)/(2*pi^3)
Potencias [src] 
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         /                                                                                                      2                                                                                                        2 
        /    /       /4*pi*k\        /2*pi*k\        /2*pi*k\        /4*pi*k\         /2*pi*k\         /4*pi*k\\      /       /2*pi*k\        /4*pi*k\         /2*pi*k\        /2*pi*k\        /4*pi*k\         /4*pi*k\\  
       /     |  9*cos|------|   3*sin|------|   9*cos|------|   3*sin|------|   27*sin|------|   27*sin|------||      |  3*cos|------|   9*sin|------|   27*cos|------|   9*sin|------|   3*cos|------|   27*cos|------||  
      /      |       \  3   /        \  3   /        \  3   /        \  3   /         \  3   /         \  3   /|      |       \  3   /        \  3   /         \  3   /        \  3   /        \  3   /         \  3   /|  
     /     4*|- ------------- + ------------- - ------------- - ------------- + -------------- + --------------|    4*|- ------------- + ------------- - -------------- - ------------- - ------------- + --------------|  
    /        |        2  2           pi*k              2  2         2*pi*k             3  3             3  3   |      |       pi*k             2  2             3  3             2  2         2*pi*k             3  3   |  
   /         \      pi *k                          2*pi *k                         4*pi *k          4*pi *k    /      \                      pi *k          4*pi *k          2*pi *k                         4*pi *k    /  
  /        ------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------ 
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4(3sin⁡(2πk3)πk−3sin⁡(4πk3)2πk−9cos⁡(2πk3)2π2k2−9cos⁡(4πk3)π2k2+27sin⁡(2πk3)4π3k3+27sin⁡(4πk3)4π3k3)29+4(−3cos⁡(2πk3)πk−3cos⁡(4πk3)2πk−9sin⁡(2πk3)2π2k2+9sin⁡(4πk3)π2k2−27cos⁡(2πk3)4π3k3+27cos⁡(4πk3)4π3k3)29\sqrt{\frac{4 \left(\frac{3 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{3 \sin{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k} - \frac{9 \cos{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}} - \frac{9 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{27 \sin{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}} + \frac{27 \sin{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9} + \frac{4 \left(- \frac{3 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{3 \cos{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k} - \frac{9 \sin{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}} + \frac{9 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{27 \cos{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}} + \frac{27 \cos{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9}} 
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            /                                                                                                                                                                                              2                                                                                                                                                                                                      2 
           /    /     -2*pi*I*k      2*pi*I*k      -4*pi*I*k      4*pi*I*k       -2*pi*I*k       2*pi*I*k       -4*pi*I*k       4*pi*I*k                                                                  \      /     -4*pi*I*k      4*pi*I*k      -2*pi*I*k      2*pi*I*k                                                                                                                                      \  
          /     |     ---------      --------      ---------      --------       ---------       --------       ---------       --------                                                                  |      |     ---------      --------      ---------      --------                                                                                                                                      |  
         /      |         3             3              3             3               3              3               3              3           /   -4*pi*I*k    4*pi*I*k\       /   -2*pi*I*k    2*pi*I*k\|      |         3             3              3             3            /   -4*pi*I*k    4*pi*I*k\        /   -2*pi*I*k    2*pi*I*k\       /   -2*pi*I*k    2*pi*I*k\       /   -4*pi*I*k    4*pi*I*k\|  
        /       |  3*e            3*e           3*e            3*e           27*e            27*e           27*e            27*e               |   ---------    --------|       |   ---------    --------||      |  9*e            9*e           9*e            9*e                |   ---------    --------|        |   ---------    --------|       |   ---------    --------|       |   ---------    --------||  
       /        |- ------------ - -----------   ------------ + -----------   ------------- + ------------   ------------- + ------------       |       3           3    |       |       3           3    ||      |- ------------ - -----------   ------------ + -----------        |       3           3    |        |       3           3    |       |       3           3    |       |       3           3    ||  
      /         |       2              2             2              2              2              2               2              2         9*I*\- e          + e        /   9*I*\- e          + e        /|      |       2              2             2              2        27*I*\- e          + e        /   27*I*\- e          + e        /   3*I*\- e          + e        /   3*I*\- e          + e        /|  
     /        4*|---------------------------- - -------------------------- - ---------------------------- + ---------------------------- - ------------------------------ + ------------------------------|    4*|---------------------------- - -------------------------- - ------------------------------- - ------------------------------- - ------------------------------ + ------------------------------|  
    /           |            pi*k                         2*pi*k                           3  3                           3  3                            2  2                             2  2           |      |             2  2                           2  2                            3  3                              3  3                          2*pi*k                           4*pi*k            |  
   /            \                                                                      4*pi *k                        4*pi *k                         2*pi *k                          4*pi *k            /      \           pi *k                        2*pi *k                         8*pi *k                           8*pi *k                                                                              /  
  /           ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 
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4(−3e2iπk32−3e−2iπk32πk−3e4iπk32+3e−4iπk322πk+9i(e2iπk3−e−2iπk3)4π2k2−9i(e4iπk3−e−4iπk3)2π2k2−27e2iπk32+27e−2iπk324π3k3+27e4iπk32+27e−4iπk324π3k3)29+4(−3i(e2iπk3−e−2iπk3)2πk+3i(e4iπk3−e−4iπk3)4πk−9e2iπk32+9e−2iπk322π2k2+−9e4iπk32−9e−4iπk32π2k2−27i(e2iπk3−e−2iπk3)8π3k3−27i(e4iπk3−e−4iπk3)8π3k3)29\sqrt{\frac{4 \left(\frac{- \frac{3 e^{\frac{2 i \pi k}{3}}}{2} - \frac{3 e^{- \frac{2 i \pi k}{3}}}{2}}{\pi k} - \frac{\frac{3 e^{\frac{4 i \pi k}{3}}}{2} + \frac{3 e^{- \frac{4 i \pi k}{3}}}{2}}{2 \pi k} + \frac{9 i \left(e^{\frac{2 i \pi k}{3}} - e^{- \frac{2 i \pi k}{3}}\right)}{4 \pi^{2} k^{2}} - \frac{9 i \left(e^{\frac{4 i \pi k}{3}} - e^{- \frac{4 i \pi k}{3}}\right)}{2 \pi^{2} k^{2}} - \frac{\frac{27 e^{\frac{2 i \pi k}{3}}}{2} + \frac{27 e^{- \frac{2 i \pi k}{3}}}{2}}{4 \pi^{3} k^{3}} + \frac{\frac{27 e^{\frac{4 i \pi k}{3}}}{2} + \frac{27 e^{- \frac{4 i \pi k}{3}}}{2}}{4 \pi^{3} k^{3}}\right)^{2}}{9} + \frac{4 \left(- \frac{3 i \left(e^{\frac{2 i \pi k}{3}} - e^{- \frac{2 i \pi k}{3}}\right)}{2 \pi k} + \frac{3 i \left(e^{\frac{4 i \pi k}{3}} - e^{- \frac{4 i \pi k}{3}}\right)}{4 \pi k} - \frac{\frac{9 e^{\frac{2 i \pi k}{3}}}{2} + \frac{9 e^{- \frac{2 i \pi k}{3}}}{2}}{2 \pi^{2} k^{2}} + \frac{- \frac{9 e^{\frac{4 i \pi k}{3}}}{2} - \frac{9 e^{- \frac{4 i \pi k}{3}}}{2}}{\pi^{2} k^{2}} - \frac{27 i \left(e^{\frac{2 i \pi k}{3}} - e^{- \frac{2 i \pi k}{3}}\right)}{8 \pi^{3} k^{3}} - \frac{27 i \left(e^{\frac{4 i \pi k}{3}} - e^{- \frac{4 i \pi k}{3}}\right)}{8 \pi^{3} k^{3}}\right)^{2}}{9}} 
sqrt(4*((-3*exp(-2*pi*i*k/3)/2 - 3*exp(2*pi*i*k/3)/2)/(pi*k) - (3*exp(-4*pi*i*k/3)/2 + 3*exp(4*pi*i*k/3)/2)/(2*pi*k) - (27*exp(-2*pi*i*k/3)/2 + 27*exp(2*pi*i*k/3)/2)/(4*pi^3*k^3) + (27*exp(-4*pi*i*k/3)/2 + 27*exp(4*pi*i*k/3)/2)/(4*pi^3*k^3) - 9*i*(-exp(-4*pi*i*k/3) + exp(4*pi*i*k/3))/(2*pi^2*k^2) + 9*i*(-exp(-2*pi*i*k/3) + exp(2*pi*i*k/3))/(4*pi^2*k^2))^2/9 + 4*((-9*exp(-4*pi*i*k/3)/2 - 9*exp(4*pi*i*k/3)/2)/(pi^2*k^2) - (9*exp(-2*pi*i*k/3)/2 + 9*exp(2*pi*i*k/3)/2)/(2*pi^2*k^2) - 27*i*(-exp(-4*pi*i*k/3) + exp(4*pi*i*k/3))/(8*pi^3*k^3) - 27*i*(-exp(-2*pi*i*k/3) + exp(2*pi*i*k/3))/(8*pi^3*k^3) - 3*i*(-exp(-2*pi*i*k/3) + exp(2*pi*i*k/3))/(2*pi*k) + 3*i*(-exp(-4*pi*i*k/3) + exp(4*pi*i*k/3))/(4*pi*k))^2/9)
Abrimos la expresión [src] 
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         /                                                                                                      2                                                                                                        2 
        /    /       /4*pi*k\        /2*pi*k\        /2*pi*k\        /4*pi*k\         /2*pi*k\         /4*pi*k\\      /       /2*pi*k\        /4*pi*k\         /2*pi*k\        /2*pi*k\        /4*pi*k\         /4*pi*k\\  
       /     |  9*cos|------|   3*sin|------|   9*cos|------|   3*sin|------|   27*sin|------|   27*sin|------||      |  3*cos|------|   9*sin|------|   27*cos|------|   9*sin|------|   3*cos|------|   27*cos|------||  
      /      |       \  3   /        \  3   /        \  3   /        \  3   /         \  3   /         \  3   /|      |       \  3   /        \  3   /         \  3   /        \  3   /        \  3   /         \  3   /|  
     /     4*|- ------------- + ------------- - ------------- - ------------- + -------------- + --------------|    4*|- ------------- + ------------- - -------------- - ------------- - ------------- + --------------|  
    /        |        2  2           pi*k              2  2         2*pi*k             3  3             3  3   |      |       pi*k             2  2             3  3             2  2         2*pi*k             3  3   |  
   /         \      pi *k                          2*pi *k                         4*pi *k          4*pi *k    /      \                      pi *k          4*pi *k          2*pi *k                         4*pi *k    /  
  /        ------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------ 
\/                                                           9                                                                                                        9
4(3sin⁡(2πk3)πk−3sin⁡(4πk3)2πk−9cos⁡(2πk3)2π2k2−9cos⁡(4πk3)π2k2+27sin⁡(2πk3)4π3k3+27sin⁡(4πk3)4π3k3)29+4(−3cos⁡(2πk3)πk−3cos⁡(4πk3)2πk−9sin⁡(2πk3)2π2k2+9sin⁡(4πk3)π2k2−27cos⁡(2πk3)4π3k3+27cos⁡(4πk3)4π3k3)29\sqrt{\frac{4 \left(\frac{3 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{3 \sin{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k} - \frac{9 \cos{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}} - \frac{9 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{27 \sin{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}} + \frac{27 \sin{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9} + \frac{4 \left(- \frac{3 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{3 \cos{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k} - \frac{9 \sin{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}} + \frac{9 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{27 \cos{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}} + \frac{27 \cos{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9}} 
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      /    2/4*pi*k\      2/4*pi*k\        2/2*pi*k\        2/2*pi*k\         2/2*pi*k\         2/4*pi*k\         2/2*pi*k\         2/4*pi*k\         2/2*pi*k\         2/4*pi*k\         2/2*pi*k\         2/4*pi*k\         /2*pi*k\    /4*pi*k\         /4*pi*k\    /2*pi*k\         /2*pi*k\    /4*pi*k\         /2*pi*k\    /4*pi*k\         /4*pi*k\    /2*pi*k\        /2*pi*k\    /4*pi*k\        /2*pi*k\    /4*pi*k\         /2*pi*k\    /4*pi*k\         /2*pi*k\    /4*pi*k\         /2*pi*k\    /4*pi*k\ 
     /  cos |------|   sin |------|   4*cos |------|   4*sin |------|   27*cos |------|   27*cos |------|   27*sin |------|   27*sin |------|   81*cos |------|   81*cos |------|   81*sin |------|   81*sin |------|   81*cos|------|*sin|------|   81*cos|------|*sin|------|   27*sin|------|*sin|------|   18*cos|------|*sin|------|   18*cos|------|*sin|------|   4*sin|------|*sin|------|   4*cos|------|*cos|------|   27*cos|------|*cos|------|   81*cos|------|*cos|------|   81*sin|------|*sin|------| 
    /       \  3   /       \  3   /         \  3   /         \  3   /          \  3   /          \  3   /          \  3   /          \  3   /          \  3   /          \  3   /          \  3   /          \  3   /         \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   /        \  3   /    \  3   /        \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   /         \  3   /    \  3   / 
   /    ------------ + ------------ + -------------- + -------------- + --------------- + --------------- + --------------- + --------------- + --------------- + --------------- + --------------- + --------------- - -------------------------- - -------------------------- - -------------------------- - -------------------------- - -------------------------- - ------------------------- + ------------------------- + -------------------------- - -------------------------- + -------------------------- 
  /          2  2           2  2            2  2             2  2              4  4              4  4              4  4              4  4               6  6              6  6              6  6              6  6                  5  5                         5  5                         4  4                         3  3                         3  3                         2  2                        2  2                        4  4                          6  6                         6  6          
\/         pi *k          pi *k           pi *k            pi *k             pi *k             pi *k             pi *k             pi *k            4*pi *k           4*pi *k           4*pi *k           4*pi *k                 pi *k                        pi *k                        pi *k                        pi *k                        pi *k                        pi *k                       pi *k                       pi *k                       2*pi *k                      2*pi *k
4sin⁡2(2πk3)π2k2−4sin⁡(2πk3)sin⁡(4πk3)π2k2+sin⁡2(4πk3)π2k2+4cos⁡2(2πk3)π2k2+4cos⁡(2πk3)cos⁡(4πk3)π2k2+cos⁡2(4πk3)π2k2−18sin⁡(2πk3)cos⁡(4πk3)π3k3−18sin⁡(4πk3)cos⁡(2πk3)π3k3+27sin⁡2(2πk3)π4k4−27sin⁡(2πk3)sin⁡(4πk3)π4k4+27sin⁡2(4πk3)π4k4+27cos⁡2(2πk3)π4k4+27cos⁡(2πk3)cos⁡(4πk3)π4k4+27cos⁡2(4πk3)π4k4−81sin⁡(2πk3)cos⁡(4πk3)π5k5−81sin⁡(4πk3)cos⁡(2πk3)π5k5+81sin⁡2(2πk3)4π6k6+81sin⁡(2πk3)sin⁡(4πk3)2π6k6+81sin⁡2(4πk3)4π6k6+81cos⁡2(2πk3)4π6k6−81cos⁡(2πk3)cos⁡(4πk3)2π6k6+81cos⁡2(4πk3)4π6k6\sqrt{\frac{4 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{4 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{\sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{4 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{4 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{\cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{18 \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{3} k^{3}} - \frac{18 \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{3} k^{3}} + \frac{27 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{4} k^{4}} - \frac{27 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} - \frac{81 \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{5} k^{5}} - \frac{81 \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{5} k^{5}} + \frac{81 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{6} k^{6}} + \frac{81 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{2 \pi^{6} k^{6}} + \frac{81 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{6} k^{6}} + \frac{81 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{6} k^{6}} - \frac{81 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{2 \pi^{6} k^{6}} + \frac{81 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{6} k^{6}}} 
sqrt(cos(4*pi*k/3)^2/(pi^2*k^2) + sin(4*pi*k/3)^2/(pi^2*k^2) + 4*cos(2*pi*k/3)^2/(pi^2*k^2) + 4*sin(2*pi*k/3)^2/(pi^2*k^2) + 27*cos(2*pi*k/3)^2/(pi^4*k^4) + 27*cos(4*pi*k/3)^2/(pi^4*k^4) + 27*sin(2*pi*k/3)^2/(pi^4*k^4) + 27*sin(4*pi*k/3)^2/(pi^4*k^4) + 81*cos(2*pi*k/3)^2/(4*pi^6*k^6) + 81*cos(4*pi*k/3)^2/(4*pi^6*k^6) + 81*sin(2*pi*k/3)^2/(4*pi^6*k^6) + 81*sin(4*pi*k/3)^2/(4*pi^6*k^6) - 81*cos(2*pi*k/3)*sin(4*pi*k/3)/(pi^5*k^5) - 81*cos(4*pi*k/3)*sin(2*pi*k/3)/(pi^5*k^5) - 27*sin(2*pi*k/3)*sin(4*pi*k/3)/(pi^4*k^4) - 18*cos(2*pi*k/3)*sin(4*pi*k/3)/(pi^3*k^3) - 18*cos(4*pi*k/3)*sin(2*pi*k/3)/(pi^3*k^3) - 4*sin(2*pi*k/3)*sin(4*pi*k/3)/(pi^2*k^2) + 4*cos(2*pi*k/3)*cos(4*pi*k/3)/(pi^2*k^2) + 27*cos(2*pi*k/3)*cos(4*pi*k/3)/(pi^4*k^4) - 81*cos(2*pi*k/3)*cos(4*pi*k/3)/(2*pi^6*k^6) + 81*sin(2*pi*k/3)*sin(4*pi*k/3)/(2*pi^6*k^6))
Compilar la expresión [src] 
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         /                                                                                                      2                                                                                                        2 
        /    /       /4*pi*k\        /2*pi*k\        /2*pi*k\        /4*pi*k\         /2*pi*k\         /4*pi*k\\      /       /2*pi*k\        /4*pi*k\         /2*pi*k\        /2*pi*k\        /4*pi*k\         /4*pi*k\\  
       /     |  9*cos|------|   3*sin|------|   9*cos|------|   3*sin|------|   27*sin|------|   27*sin|------||      |  3*cos|------|   9*sin|------|   27*cos|------|   9*sin|------|   3*cos|------|   27*cos|------||  
      /      |       \  3   /        \  3   /        \  3   /        \  3   /         \  3   /         \  3   /|      |       \  3   /        \  3   /         \  3   /        \  3   /        \  3   /         \  3   /|  
     /     4*|- ------------- + ------------- - ------------- - ------------- + -------------- + --------------|    4*|- ------------- + ------------- - -------------- - ------------- - ------------- + --------------|  
    /        |        2  2           pi*k              2  2         2*pi*k             3  3             3  3   |      |       pi*k             2  2             3  3             2  2         2*pi*k             3  3   |  
   /         \      pi *k                          2*pi *k                         4*pi *k          4*pi *k    /      \                      pi *k          4*pi *k          2*pi *k                         4*pi *k    /  
  /        ------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------ 
\/                                                           9                                                                                                        9
4(3sin⁡(2πk3)πk−3sin⁡(4πk3)2πk−9cos⁡(2πk3)2π2k2−9cos⁡(4πk3)π2k2+27sin⁡(2πk3)4π3k3+27sin⁡(4πk3)4π3k3)29+4(−3cos⁡(2πk3)πk−3cos⁡(4πk3)2πk−9sin⁡(2πk3)2π2k2+9sin⁡(4πk3)π2k2−27cos⁡(2πk3)4π3k3+27cos⁡(4πk3)4π3k3)29\sqrt{\frac{4 \left(\frac{3 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{3 \sin{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k} - \frac{9 \cos{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}} - \frac{9 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{27 \sin{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}} + \frac{27 \sin{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9} + \frac{4 \left(- \frac{3 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{3 \cos{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k} - \frac{9 \sin{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}} + \frac{9 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{27 \cos{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}} + \frac{27 \cos{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9}} 
sqrt(4*(-9*cos(((4*pi)*k)/3)/(pi^2*k^2) + 3*sin(((2*pi)*k)/3)/(pi*k) - 9*cos(((2*pi)*k)/3)/(2*pi^2*k^2) - 3*sin(((4*pi)*k)/3)/(2*pi*k) + 27*sin(((2*pi)*k)/3)/(4*pi^3*k^3) + 27*sin(((4*pi)*k)/3)/(4*pi^3*k^3))^2/9 + 4*(-3*cos(((2*pi)*k)/3)/(pi*k) + 9*sin(((4*pi)*k)/3)/(pi^2*k^2) - 27*cos(((2*pi)*k)/3)/(4*pi^3*k^3) - 9*sin(((2*pi)*k)/3)/(2*pi^2*k^2) - 3*cos(((4*pi)*k)/3)/(2*pi*k) + 27*cos(((4*pi)*k)/3)/(4*pi^3*k^3))^2/9)
Denominador racional [src] 
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       /                                                                                                   2                                                                                                     2 
      /  /        /4*pi*k\        /2*pi*k\        /4*pi*k\        /2*pi*k\        /2*pi*k\        /4*pi*k\\    /        /4*pi*k\        /4*pi*k\        /4*pi*k\        /2*pi*k\        /2*pi*k\        /2*pi*k\\  
     /   |  12*cos|------|   6*cos|------|   2*sin|------|   4*sin|------|   9*sin|------|   9*sin|------||    |  12*sin|------|   9*cos|------|   2*cos|------|   4*cos|------|   6*sin|------|   9*cos|------||  
    /    |        \  3   /        \  3   /        \  3   /        \  3   /        \  3   /        \  3   /|    |        \  3   /        \  3   /        \  3   /        \  3   /        \  3   /        \  3   /|  
   /     |- -------------- - ------------- - ------------- + ------------- + ------------- + -------------|  + |- -------------- - ------------- + ------------- + ------------- + ------------- + -------------|  
  /      |        2  2             2  2           pi*k            pi*k             3  3            3  3   |    |        2  2             3  3           pi*k            pi*k             2  2            3  3   |  
\/       \      pi *k            pi *k                                           pi *k           pi *k    /    \      pi *k            pi *k                                           pi *k           pi *k    /  
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                                                                                                         2
(4sin⁡(2πk3)πk−2sin⁡(4πk3)πk−6cos⁡(2πk3)π2k2−12cos⁡(4πk3)π2k2+9sin⁡(2πk3)π3k3+9sin⁡(4πk3)π3k3)2+(4cos⁡(2πk3)πk+2cos⁡(4πk3)πk+6sin⁡(2πk3)π2k2−12sin⁡(4πk3)π2k2+9cos⁡(2πk3)π3k3−9cos⁡(4πk3)π3k3)22\frac{\sqrt{\left(\frac{4 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{2 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi k} - \frac{6 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{12 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{9 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi^{3} k^{3}} + \frac{9 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{3} k^{3}}\right)^{2} + \left(\frac{4 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k} + \frac{2 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi k} + \frac{6 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{12 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{9 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{3} k^{3}} - \frac{9 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{3} k^{3}}\right)^{2}}}{2} 
sqrt((-12*cos(4*pi*k/3)/(pi^2*k^2) - 6*cos(2*pi*k/3)/(pi^2*k^2) - 2*sin(4*pi*k/3)/(pi*k) + 4*sin(2*pi*k/3)/(pi*k) + 9*sin(2*pi*k/3)/(pi^3*k^3) + 9*sin(4*pi*k/3)/(pi^3*k^3))^2 + (-12*sin(4*pi*k/3)/(pi^2*k^2) - 9*cos(4*pi*k/3)/(pi^3*k^3) + 2*cos(4*pi*k/3)/(pi*k) + 4*cos(2*pi*k/3)/(pi*k) + 6*sin(2*pi*k/3)/(pi^2*k^2) + 9*cos(2*pi*k/3)/(pi^3*k^3))^2)/2
    © Sr Examen – Калькулятор