# set terminal pngcairo  transparent enhanced font "arial,10" fontscale 1.0 size 600, 400 
# set output 'bivariat.8.png'
set key fixed right bottom vertical Right noreverse enhanced autotitle nobox
set samples 500, 500
set style data lines
set title "Finite summation of 10, 100, 1000 fourier coefficients" 
set xrange [ -10.0000 : 10.0000 ] noreverse
set yrange [ -0.400000 : 1.20000 ] noreverse
integral_f(x) = (x>0)?int1a(x,x/ceil(x/delta)):-int1b(x,-x/ceil(-x/delta))
int1a(x,d) = (x<=d*.1) ? 0 : (int1a(x-d,d)+(f(x-d)+4*f(x-d*.5)+f(x))*d/6.)
int1b(x,d) = (x>=-d*.1) ? 0 : (int1b(x+d,d)+(f(x+d)+4*f(x+d*.5)+f(x))*d/6.)
f(x)=sin(x-1)-.75*sin(2*x-1)+(x**2)/8-5
integral2_f(x,y) = (x<y)?int2(x,y,(y-x)/ceil((y-x)/delta)):                         -int2(y,x,(x-y)/ceil((x-y)/delta))
int2(x,y,d) = (x>y-d*.5) ? 0 : (int2(x+d,y,d) + (f(x)+4*f(x+d*.5)+f(x+d))*d/6.)
ack(m,n) = (m == 0) ? n + 1 : (n == 0) ? ack(m-1,1) : ack(m-1,ack(m,n-1))
min(x,y) = (x < y) ? x : y
max(x,y) = (x > y) ? x : y
gcd(x,y) = gcd1(rnd(max(x,y)),rnd(min(x,y)))
rnd(x) = int(x+0.5)
gcd1(x,y) = (y == 0) ? x : gcd1(y, x - x/y * y)
fourier(k, x) = sin(3./2*k)/k * 2./3*cos(k*x)
sum10(x)   = 1./2 + sum [k=1:10]   fourier(k, x)
sum100(x)  = 1./2 + sum [k=1:100]  fourier(k, x)
sum1000(x) = 1./2 + sum [k=1:1000] fourier(k, x)
NO_ANIMATION = 1
delta = 0.2
plot     sum10(x)   title "1./2 + sum [k=1:10]   sin(3./2*k)/k * 2./3*cos(k*x)",     sum100(x)  title "1./2 + sum [k=1:100]  sin(3./2*k)/k * 2./3*cos(k*x)",     sum1000(x) title "1./2 + sum [k=1:1000] sin(3./2*k)/k * 2./3*cos(k*x)"