@@ -1005,7 +1005,11 @@ def _derivative_(self, u, m, diff_param):
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sage: elliptic_eu(x,m).diff(x)
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sqrt(-m*jacobi_sn(x, m)^2 + 1)*jacobi_dn(x, m)
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sage: elliptic_eu(x,m).diff(m)
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- 1/2*(elliptic_eu(x, m) - elliptic_f(jacobi_am(x, m), m))/m - 1/2*(m*jacobi_cn(x, m)*jacobi_sn(x, m) - (m - 1)*x - elliptic_eu(x, m)*jacobi_dn(x, m))*sqrt(-m*jacobi_sn(x, m)^2 + 1)/((m - 1)*m)
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+ 1/2*(elliptic_eu(x, m)
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+ - elliptic_f(jacobi_am(x, m), m))/m
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+ - 1/2*(m*jacobi_cn(x, m)*jacobi_sn(x, m)
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+ - (m - 1)*x
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+ - elliptic_eu(x, m)*jacobi_dn(x, m))*sqrt(-m*jacobi_sn(x, m)^2 + 1)/((m - 1)*m)
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"""
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from sage .functions .jacobi import jacobi , jacobi_am
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if diff_param == 0 :
@@ -1143,7 +1147,9 @@ def _derivative_(self, z, m, diff_param):
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sage: elliptic_f(x,m).diff(x)
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1/sqrt(-m*sin(x)^2 + 1)
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sage: elliptic_f(x,m).diff(m)
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- -1/2*elliptic_f(x, m)/m + 1/4*sin(2*x)/(sqrt(-m*sin(x)^2 + 1)*(m - 1)) - 1/2*elliptic_e(x, m)/((m - 1)*m)
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+ -1/2*elliptic_f(x, m)/m
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+ + 1/4*sin(2*x)/(sqrt(-m*sin(x)^2 + 1)*(m - 1))
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+ - 1/2*elliptic_e(x, m)/((m - 1)*m)
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"""
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if diff_param == 0 :
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return Integer (1 ) / sqrt (Integer (1 ) - m * sin (z ) ** Integer (2 ))
@@ -1238,7 +1244,8 @@ def _derivative_(self, z, diff_param):
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EXAMPLES::
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sage: elliptic_kc(x).diff(x)
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- -1/2*((x - 1)*elliptic_kc(x) + elliptic_ec(x))/((x - 1)*x)
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+ -1/2*((x - 1)*elliptic_kc(x)
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+ + elliptic_ec(x))/((x - 1)*x)
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"""
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return ((elliptic_ec (z ) - (Integer (1 ) - z ) * elliptic_kc (z )) /
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(Integer (2 ) * (Integer (1 ) - z ) * z ))
@@ -1327,11 +1334,16 @@ def _derivative_(self, n, z, m, diff_param):
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sage: n,z,m = var('n,z,m')
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sage: elliptic_pi(n,z,m).diff(n)
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- 1/4*(sqrt(-m*sin(z)^2 + 1)*n*sin(2*z)/(n*sin(z)^2 - 1) + 2*(m - n)*elliptic_f(z, m)/n + 2*(n^2 - m)*elliptic_pi(n, z, m)/n + 2*elliptic_e(z, m))/((m - n)*(n - 1))
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+ 1/4*(sqrt(-m*sin(z)^2 + 1)*n*sin(2*z)/(n*sin(z)^2 - 1)
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+ + 2*(m - n)*elliptic_f(z, m)/n
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+ + 2*(n^2 - m)*elliptic_pi(n, z, m)/n
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+ + 2*elliptic_e(z, m))/((m - n)*(n - 1))
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sage: elliptic_pi(n,z,m).diff(z)
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-1/(sqrt(-m*sin(z)^2 + 1)*(n*sin(z)^2 - 1))
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sage: elliptic_pi(n,z,m).diff(m)
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- 1/4*(m*sin(2*z)/(sqrt(-m*sin(z)^2 + 1)*(m - 1)) - 2*elliptic_e(z, m)/(m - 1) - 2*elliptic_pi(n, z, m))/(m - n)
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+ 1/4*(m*sin(2*z)/(sqrt(-m*sin(z)^2 + 1)*(m - 1))
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+ - 2*elliptic_e(z, m)/(m - 1)
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+ - 2*elliptic_pi(n, z, m))/(m - n)
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"""
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if diff_param == 0 :
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return ((Integer (1 ) / (Integer (2 ) * (m - n ) * (n - Integer (1 )))) *
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