#
# Elliptic integrals of the third kind
#
save_encoding = GPVAL_ENCODING
set encoding utf8

set title "Complete elliptic integral of the third kind EllipticPi(n,k)"

set key reverse left Left
set key title "Π(n,k) = {/*2 ∫@_{/*0.5 0}^{/*.5 π/2}}"\
             ."{/:Italic [1-nsin^2(t)]^{-1} sqrt(1-k^2sin^2(t))^{-1} dt}\n"

set border 3
set margins 10,10,-1,4
set sample 250
set xrange [0:1]
set xtics 0.1 format "%.1f" nomirror
set yrange [0:6]
set ytics 1 nomirror
set xlabel "{/:Italic k}"

plot EllipticPi(0.5,x) with lines lw 2 title "n = 0.5", \
     EllipticPi(0.0,x) with lines lw 2 title "n = 0", \
     EllipticPi(-1.,x) with lines lw 2 title "n = -1", \
     EllipticPi(-3.,x) with lines lw 2 title "n = -3", \
     EllipticPi(-10.,x) with lines lw 2 lc "brown" title "n = -10"


Click here for minimal script to generate this plot

set title "Complete elliptic integrals of the first and second kinds"
set key notitle invert top center spacing 2
set xtics 0.5
set xlabel "x"
set xrange [-1:1]
set yrange [1:3]

plot EllipticE(x) lw 2 title "\nEllipticE(x)={/*1.5 ∫@_{/*0.5 0}^{/*.5 π/2}}{/:Italic sqrt(1-k^2sin^2(t)) dt}\n",\
     EllipticK(x) lw 2 title "\nEllipticK(x)={/*1.5 ∫@_{/*0.5 0}^{/*.5 π/2}}{/:Italic sqrt(1-k^2sin^2(t))^{-1} dt}\n"


Click here for minimal script to generate this plot

set encoding save_encoding