Sr Examen
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?
Solución
Ha introducido
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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
$$\sqrt{\frac{4 \left(\left(\left(\left(\left(\frac{9 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{\left(-1\right) 3 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k}\right) - \frac{27 \cos{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right) - \frac{9 \sin{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}}\right) - \frac{3 \cos{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k}\right) + \frac{27 \cos{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9} + \frac{4 \left(\left(\left(\left(\left(\frac{\left(-1\right) 9 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{3 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi k}\right) - \frac{9 \cos{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}}\right) - \frac{3 \sin{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k}\right) + \frac{27 \sin{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right) + \frac{27 \sin{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9}}$$
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
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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))
$$\sqrt{\frac{4 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{4 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{\sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{4 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{4 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{\cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{18 \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{3} k^{3}} - \frac{18 \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{3} k^{3}} + \frac{27 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{4} k^{4}} - \frac{27 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} + \frac{27 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{4}} - \frac{81 \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{5} k^{5}} - \frac{81 \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{5} k^{5}} + \frac{81 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{6} k^{6}} + \frac{81 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{2 \pi^{6} k^{6}} + \frac{81 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{6} k^{6}} + \frac{81 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{6} k^{6}} - \frac{81 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{2 \pi^{6} k^{6}} + \frac{81 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{6} k^{6}}}$$
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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
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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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/ 6
\/ k
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3
2*pi
$$\frac{\sqrt{\frac{\left(2 \pi^{2} k^{2} \left(2 \cos{\left(\frac{2 \pi k}{3} \right)} + \cos{\left(\frac{4 \pi k}{3} \right)}\right) + 6 \pi k \left(\sin{\left(\frac{2 \pi k}{3} \right)} - 2 \sin{\left(\frac{4 \pi k}{3} \right)}\right) + 9 \cos{\left(\frac{2 \pi k}{3} \right)} - 9 \cos{\left(\frac{4 \pi k}{3} \right)}\right)^{2} + \left(16 \pi^{2} k^{2} \sin^{3}{\left(\frac{\pi k}{3} \right)} \cos{\left(\frac{\pi k}{3} \right)} - 6 \pi k \left(\cos{\left(\frac{2 \pi k}{3} \right)} + 2 \cos{\left(\frac{4 \pi k}{3} \right)}\right) + 9 \sin{\left(\frac{2 \pi k}{3} \right)} + 9 \sin{\left(\frac{4 \pi k}{3} \right)}\right)^{2}}{k^{6}}}}{2 \pi^{3}}$$
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
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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
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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
$$\frac{\sqrt{\frac{16 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{2}} - \frac{16 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{4 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{16 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{2}} + \frac{16 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{4 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} - \frac{72 \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi k^{3}} - \frac{72 \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k^{3}} + \frac{108 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{4}} - \frac{108 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{4}} + \frac{108 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{4}} + \frac{108 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{4}} + \frac{108 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{4}} + \frac{108 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{4}} - \frac{324 \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{3} k^{5}} - \frac{324 \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{3} k^{5}} + \frac{81 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{4} k^{6}} + \frac{162 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{6}} + \frac{81 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{6}} + \frac{81 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{\pi^{4} k^{6}} - \frac{162 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{6}} + \frac{81 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{\pi^{4} k^{6}}}}{2 \pi}$$
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
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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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/ 6
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3
2*pi
$$\frac{\sqrt{\frac{\left(2 \pi k \left(2 \pi k \sin{\left(\frac{2 \pi k}{3} \right)} - \pi k \sin{\left(\frac{4 \pi k}{3} \right)} - 3 \cos{\left(\frac{2 \pi k}{3} \right)} - 6 \cos{\left(\frac{4 \pi k}{3} \right)}\right) + 9 \sin{\left(\frac{2 \pi k}{3} \right)} + 9 \sin{\left(\frac{4 \pi k}{3} \right)}\right)^{2} + \left(- 2 \pi^{2} k^{2} \cos{\left(\frac{4 \pi k}{3} \right)} + 4 \pi k \left(- \pi k \cos{\left(\frac{2 \pi k}{3} \right)} + 3 \sin{\left(\frac{4 \pi k}{3} \right)}\right) - 6 \pi k \sin{\left(\frac{2 \pi k}{3} \right)} - 9 \cos{\left(\frac{2 \pi k}{3} \right)} + 9 \cos{\left(\frac{4 \pi k}{3} \right)}\right)^{2}}{k^{6}}}}{2 \pi^{3}}$$
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
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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
/ k k k k k k k k k k k k k k
/ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ 2
\/ k
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3
2*pi
$$\frac{\sqrt{\frac{16 \pi^{4} \sin^{2}{\left(\frac{2 \pi k}{3} \right)} - 16 \pi^{4} \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)} + 4 \pi^{4} \sin^{2}{\left(\frac{4 \pi k}{3} \right)} + 16 \pi^{4} \cos^{2}{\left(\frac{2 \pi k}{3} \right)} + 16 \pi^{4} \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)} + 4 \pi^{4} \cos^{2}{\left(\frac{4 \pi k}{3} \right)} - \frac{72 \pi^{3} \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{k} - \frac{72 \pi^{3} \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{k} + \frac{108 \pi^{2} \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{2}} - \frac{108 \pi^{2} \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{108 \pi^{2} \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{108 \pi^{2} \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{2}} + \frac{108 \pi^{2} \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} + \frac{108 \pi^{2} \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{2}} - \frac{324 \pi \sin{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{k^{3}} - \frac{324 \pi \sin{\left(\frac{4 \pi k}{3} \right)} \cos{\left(\frac{2 \pi k}{3} \right)}}{k^{3}} + \frac{81 \sin^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{4}} + \frac{162 \sin{\left(\frac{2 \pi k}{3} \right)} \sin{\left(\frac{4 \pi k}{3} \right)}}{k^{4}} + \frac{81 \sin^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{4}} + \frac{81 \cos^{2}{\left(\frac{2 \pi k}{3} \right)}}{k^{4}} - \frac{162 \cos{\left(\frac{2 \pi k}{3} \right)} \cos{\left(\frac{4 \pi k}{3} \right)}}{k^{4}} + \frac{81 \cos^{2}{\left(\frac{4 \pi k}{3} \right)}}{k^{4}}}{k^{2}}}}{2 \pi^{3}}$$
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 /
/ ------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------
\/ 9 9
$$\sqrt{\frac{4 \left(\frac{3 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{3 \sin{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k} - \frac{9 \cos{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}} - \frac{9 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{27 \sin{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}} + \frac{27 \sin{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9} + \frac{4 \left(- \frac{3 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{3 \cos{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k} - \frac{9 \sin{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}} + \frac{9 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{27 \cos{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}} + \frac{27 \cos{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9}}$$
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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 /
/ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\/ 9 9
$$\sqrt{\frac{4 \left(\frac{- \frac{3 e^{\frac{2 i \pi k}{3}}}{2} - \frac{3 e^{- \frac{2 i \pi k}{3}}}{2}}{\pi k} - \frac{\frac{3 e^{\frac{4 i \pi k}{3}}}{2} + \frac{3 e^{- \frac{4 i \pi k}{3}}}{2}}{2 \pi k} + \frac{9 i \left(e^{\frac{2 i \pi k}{3}} - e^{- \frac{2 i \pi k}{3}}\right)}{4 \pi^{2} k^{2}} - \frac{9 i \left(e^{\frac{4 i \pi k}{3}} - e^{- \frac{4 i \pi k}{3}}\right)}{2 \pi^{2} k^{2}} - \frac{\frac{27 e^{\frac{2 i \pi k}{3}}}{2} + \frac{27 e^{- \frac{2 i \pi k}{3}}}{2}}{4 \pi^{3} k^{3}} + \frac{\frac{27 e^{\frac{4 i \pi k}{3}}}{2} + \frac{27 e^{- \frac{4 i \pi k}{3}}}{2}}{4 \pi^{3} k^{3}}\right)^{2}}{9} + \frac{4 \left(- \frac{3 i \left(e^{\frac{2 i \pi k}{3}} - e^{- \frac{2 i \pi k}{3}}\right)}{2 \pi k} + \frac{3 i \left(e^{\frac{4 i \pi k}{3}} - e^{- \frac{4 i \pi k}{3}}\right)}{4 \pi k} - \frac{\frac{9 e^{\frac{2 i \pi k}{3}}}{2} + \frac{9 e^{- \frac{2 i \pi k}{3}}}{2}}{2 \pi^{2} k^{2}} + \frac{- \frac{9 e^{\frac{4 i \pi k}{3}}}{2} - \frac{9 e^{- \frac{4 i \pi k}{3}}}{2}}{\pi^{2} k^{2}} - \frac{27 i \left(e^{\frac{2 i \pi k}{3}} - e^{- \frac{2 i \pi k}{3}}\right)}{8 \pi^{3} k^{3}} - \frac{27 i \left(e^{\frac{4 i \pi k}{3}} - e^{- \frac{4 i \pi k}{3}}\right)}{8 \pi^{3} k^{3}}\right)^{2}}{9}}$$
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)
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
$$\sqrt{\frac{4 \left(\frac{3 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{3 \sin{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k} - \frac{9 \cos{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}} - \frac{9 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{27 \sin{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}} + \frac{27 \sin{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9} + \frac{4 \left(- \frac{3 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{3 \cos{\left(\frac{4 \pi k}{3} \right)}}{2 \pi k} - \frac{9 \sin{\left(\frac{2 \pi k}{3} \right)}}{2 \pi^{2} k^{2}} + \frac{9 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{27 \cos{\left(\frac{2 \pi k}{3} \right)}}{4 \pi^{3} k^{3}} + \frac{27 \cos{\left(\frac{4 \pi k}{3} \right)}}{4 \pi^{3} k^{3}}\right)^{2}}{9}}$$
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
$$\frac{\sqrt{\left(\frac{4 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi k} - \frac{2 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi k} - \frac{6 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{12 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{9 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi^{3} k^{3}} + \frac{9 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{3} k^{3}}\right)^{2} + \left(\frac{4 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi k} + \frac{2 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi k} + \frac{6 \sin{\left(\frac{2 \pi k}{3} \right)}}{\pi^{2} k^{2}} - \frac{12 \sin{\left(\frac{4 \pi k}{3} \right)}}{\pi^{2} k^{2}} + \frac{9 \cos{\left(\frac{2 \pi k}{3} \right)}}{\pi^{3} k^{3}} - \frac{9 \cos{\left(\frac{4 \pi k}{3} \right)}}{\pi^{3} k^{3}}\right)^{2}}}{2}$$
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