This is the parameter and formula file for generating the mandelbrot series , first and second derivatives thereof and an associated first order differential equation . For possible use with FreeBASIC , FracD1iM.bas code . To do so convert the *.gif file produced to fract003.bmp and place in the same directory as FracD1iM.bas . The FreeBASIC code should run in Windows , Linux and possibly DOS , you'll require the appropriate FreeBASIC compiler. file fract03.par test { reset=2004 type=formula formulafile=fractint.frm formulaname=d1iMandelbrot corners=-1.599499/1.008761/-0.9269799/1.029215 float=y maxiter=1024 inside=bof60 colors=@blues.map } formula , append this to fractint.frm d1iMandelbrot(XAXIS) {; Edward Montague (c) 2016 ; ; Mandelbrot series = z ; First Derivative of Mandelbrot Series = z1 ; Second Derivative of Mandelbrot Series = z2 ; ; Differential Equation is , diff(f(z),z) + 3*f(z)*z = sin(z) ; Actually w.r.t c , where c == Pixel. ; z = Pixel z1=1 z2 = 0 u = 0 ed = 0: z2 = 2*z*z2+2*z1^2 z1 = 2*z*z1+1 u = z z = z*z + Pixel ed = 3*z*Pixel + z1 - sin(u) |ed| < 10000 && |z| > 0 }