Difference between revisions of "File:MagaplotFragment.jpg"

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Importing image file
 +
Fragment of the image
  +
http://mizugadro.mydns.jp/t/index.php/File:Magaplot300.jpg
  +
  +
[[Explicit plot]] of function [[maga]] and related functions.
  +
  +
Thin green curve: $~ y=\,$ [[nori]]$(x)$ $\,=\,$ [[mori]]$\big(\sqrt{x}\big)^2\,=\,$ $\displaystyle \frac{J_0\big(L\, \sqrt{x}\big)^2}{(1\!-\!x)^2}$
  +
  +
where $J_0=\,$[[BesselJ0]], the zeroth [[Bessel function]], and $L$ is its first zero, $L=\,$[[BesselJZero]]$[0,1]$.
  +
  +
The thick blue and thick red lines represent the half of the Bessel transform of square of [[mori]] with quadratic phase;
  +
  +
[[naga]]$(x) = \displaystyle 2 \,\int_0^\infty \, $[[mori]]$(p )^2 \, \exp(\mathrm i \,x \,p^2) \, p \mathrm d p$
  +
$\, =\displaystyle \int_0^\infty \, $[[nori]]$(q) \, \exp(\mathrm i \,x \,q)\, \mathrm d q$
  +
  +
The thick black curve shows the [[maga function]],
  +
  +
$y=\,$[[maga]]$(x) \, = 1-|$[[naga]]$(x)|^2$
  +
  +
The dotted curve shows its fit,
  +
  +
$y=\mathrm{fit2}(x)=\displaystyle \frac{ x^{3/2}}{\sqrt{5-x/2+x^2/4+x^3}}$
  +
  +
The thin blue line shows deviation of this fit from the [[maga function]]; in order to mage this deviation visible, it is scaled with factor 50:
  +
  +
$y=50\big( \mathrm{fit2}(x)-\mathrm{maga}(x)\big)$
  +
  +
==[[C++]] generator of curves==
  +
  +
<poem><nomathjax><nowiki>
  +
#include<math.h>
  +
#include<stdio.h>
  +
#include <stdlib.h>
  +
#include <complex>
  +
//using namespace std;
  +
#define z_type std::complex<double>
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define RI(x) x.real(),x.imag()
  +
#define DB double
  +
#define DO(x,y) for(x=0;x<y;x++)
  +
  +
void fft(z_type *a, int N, int o) // Arry is FIRST!
  +
{z_type u,w,t; int i,j,k,l,e=1,L,p,q,m;
  +
q=N/2; p=2; for(m=1;p<N;m++) p*=2;
  +
p=N-1; z_type y=z_type(0.,M_PI*o); j=0;
  +
for(i=0;i<p;i++){ if(i<j) { t=a[j]; a[j]=a[i]; a[i]=t;}
  +
k=q; while(k<=j){ j-=k; k/=2;}
  +
j+=k; }
  +
for(l=0;l<m;l++)
  +
{ L=e; e*=2; u=1.; w=exp(y/((double)L));
  +
for(j=0;j<L;j++)
  +
{ for(i=j;i<N;i+=e){k=i+L; t=a[k]*u; a[k]=a[i]-t; a[i]+=t;}
  +
u*=w;
  +
} }
  +
}
  +
  +
void ado(FILE *O, int X, int Y)
  +
{ fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
  +
fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
  +
fprintf(O,"/M {moveto} bind def\n");
  +
fprintf(O,"/L {lineto} bind def\n");
  +
fprintf(O,"/S {stroke} bind def\n");
  +
fprintf(O,"/s {show newpath} bind def\n");
  +
fprintf(O,"/C {closepath} bind def\n");
  +
fprintf(O,"/F {fill} bind def\n");
  +
fprintf(O,"/o {.01 0 360 arc C S} bind def\n");
  +
fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
  +
fprintf(O,"/W {setlinewidth} bind def\n");
  +
fprintf(O,"/RGB {setrgbcolor} bind def\n");}
  +
  +
// #include "ado.cin"
  +
// #include"fafo.cin"
  +
//#include "mori.cin"
  +
DB L1= 2.404825557695773;
  +
DB L2= 5.5200781102863115;
  +
DB L3= 8.653727912911013;
  +
DB morin(DB x){ return j0(L1*x)/(1-x*x);}
  +
DB mori0(DB x){ int n,m; DB s, xx=x*x;
  +
DB c[16]={ 1., -0.4457964907366961303, 0.07678538241994023453, -0.0071642885058902232688,
  +
0.00042159522055140947688, -0.000017110542281627483109, 5.0832583976057607495e-7, -1.1537378620148452816e-8,
  +
2.0662789231930073316e-10, -2.9948657413756059965e-12, 3.5852738451127332173e-14,-3.6050239634659700777e-16,
  +
3.0877184831292878827e-18, -2.2798156440952688462e-20, 1.4660907878585489441e-22,-8.2852774398657968065e-25};
  +
// 16th term seems to fail; perhaps, due to the C++ rounding errors.
  +
//with m=15, at |x|<2, the error is of order of 10^(-16)
  +
//In this sense, the result is accurate while |x|<2.
  +
m=15; s=c[m]*xx; for(n=m-1;n>0;n--){ s+=c[n]; s*=xx;}
  +
return 1.+s;}
  +
DB mori(DB x){if(fabs(x)<2) return mori0(x);
  +
return morin(x);}
  +
  +
DB fit(DB x){ DB a,b,c,cc,s; a=-.055; b=0.02; c=0.45; cc=c*c;
  +
s=x*(a+x*(b+x*cc)); return c*x*sqrt(x/(1.+s)); }
  +
  +
DB fit2(DB x){ DB q; q=5.+x*(-.5+x*(.25+x)); return x*sqrt(x/q);}
  +
  +
int main(){z_type * a, *b, c; int j,m,n,N; FILE *o; DB scale,step,x,y,q,p,u,v;
  +
n=15;
  +
N=pow(2,n);
  +
//scale=.5;
  +
step=sqrt(2*M_PI/N);
  +
scale=100*step;
  +
DB dx=step*scale;
  +
DB dp=step/scale;
  +
printf("2^%2d=%8d scale=%6.4lf dx=%9.8lf dp=%9.8lf\n",n,N,scale,dx,dp);
  +
a=(z_type *) malloc((size_t)((N+1)*sizeof(z_type)));
  +
b=(z_type *) malloc((size_t)((N+1)*sizeof(z_type)));
  +
  +
DO(m,N){x=m*dx; y=mori(sqrt(x)); y*=y; a[m]=y; b[m]=y*dx; }
  +
b[0]*=.5;
  +
fft(b,N,1); //DO(j,N) printf("%2d %18.15f %18.15f %18.15f %18.15f\n", j, RI(a[j]), RI(b[j]) );
  +
//o=fopen("32.eps","w"); ado(o,1008,228);
  +
o=fopen("magaplo.eps","w"); ado(o,1008,208);
  +
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
  +
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
  +
#define o(x,y) fprintf(o,"%6.4f %6.4f o\n",0.+x,0.+y);
  +
fprintf(o,"4 104 translate 100 100 scale\n 1 setlinejoin 2 setlinecap\n");
  +
for(m=0;m<11;m++) {M(m,-1) L(m,1)}
  +
for(n=-1;n<2;n++) {M(0,n) L(10,n)} fprintf(o,".002 W S\n");
  +
  +
DO(m,N){x=dx*m; y=Re(a[m]); if(m==0)M(x,y)else L(x,y);if(x>10) break;} fprintf(o,".008 W 0 .8 0 RGB S\n");
  +
//DO(n,N){p=dp*n; y=fit(p); if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".008 W 1 0 0 RGB S\n");
  +
//DO(n,N){p=dp*n; y=fit2(p); if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".008 W 0 0 1 RGB S\n");
  +
fprintf(o,".01 W 0 0 1 RGB\n");
  +
for(n=4;n<N;n+=4){p=dp*n; y=fit2(p); o(p,y);if(p>10) break;}
  +
  +
DO(n,N){p=dp*n; y=Re(b[n]); if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".02 W 0 0 1 RGB S\n");
  +
DO(n,N){p=dp*n; y=Im(b[n]); if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".02 W 1 0 0 RGB S\n");
  +
DO(n,N){p=dp*n; u=Re(b[n]); v=Im(b[n]); y=1-u*u-v*v; if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".02 W 0 0 0 RGB S\n");
  +
// fprintf(o,".01 W 0 0 0 RGB\n");
  +
//for(n=4;n<N;n+=4){p=dp*n; u=Re(b[n]); v=Im(b[n]); y=1-u*u-v*v; o(p,y);if(p>10) break;}
  +
  +
//DO(n,N){p=dp*n; u=Re(b[n]); v=Im(b[n]); y=fit(p)-(1.-u*u-v*v); y*=50; if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".004 W .8 0 0 RGB S\n");
  +
DO(n,N){p=dp*n; u=Re(b[n]); v=Im(b[n]); y=fit2(p)-(1.-u*u-v*v); y*=50; if(n==0)M(p,y)else L(p,y);if(p>10) break;}
  +
fprintf(o,".008 W 0 0 1 RGB S\n");
  +
  +
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
  +
system("epstopdf magaplo.eps");
  +
system( "open magaplo.pdf");
  +
free(a);
  +
free(b);
  +
}
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of labels==
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry}
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
\paperwidth 420pt
  +
\paperheight 216pt
  +
\textwidth 1420pt
  +
\textheight 300pt
  +
\topmargin -108pt
  +
\oddsidemargin -73pt
  +
\newcommand \ds {\displaystyle}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rme {\mathrm{e}}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\pagestyle{empty}
  +
\parindent 0pt
  +
\begin{document}
  +
\begin{picture}(410,214)
  +
\put(10,0){\ing{magaplo}}
  +
%\put(.6,120){\sx{1.25}{$y$}}
  +
\put(2,199){\sx{1.3}{$1$}}
  +
%\put(.6,56){\sx{1.4}{$\frac{1}{2}$}}
  +
\put(2, 100){\sx{1.3}{$0$}}
  +
\put(-2, 0){\sx{1.3}{$-\!1$}}
  +
\put(111,88){\sx{1.33}{$1$}}
  +
\put(211,88){\sx{1.33}{$2$}}
  +
\put(311,89){\sx{1.33}{$3$}}
  +
\put(406,89){\sx{1.4}{$x$}}
  +
%\put(411,89){\sx{1.33}{$4$}}
  +
%\put(512,89){\sx{1.33}{$5$}}
  +
%\put(613,89){\sx{1.33}{$6$}}
  +
%\put(713,89){\sx{1.33}{$7$}}
  +
%\put(813,89){\sx{1.33}{$8$}}
  +
%\put(914,89){\sx{1.33}{$9$}}
  +
%\put(1010,89){\sx{1.4}{$x$}}
  +
%\put(15,192){\rot{-36}\sx{1.3}{$y\!=\! \mathrm{nori}(x)$}\ero}
  +
  +
\put(280,199){\rot{3}\sx{1.3}{$y\!=\! \mathrm{fit2}(x))$}\ero}
  +
\put(280,178){\rot{3}\sx{1.3}{$y\!=\! \mathrm{maga}(x))$}\ero}
  +
  +
\put(118,130){\rot{-16}\sx{1.3}{$y\!=\! \mathrm{nori}(x)$}\ero}
  +
\put(292,142){\rot{-4}\sx{1.3}{$y\!=\! \Im(\mathrm{naga}(x))$}\ero}
  +
\put(292,120){\rot{-4}\sx{1.3}{$y\!=\! \Re(\mathrm{naga}(x))$}\ero}
  +
  +
\put(170,26){\sx{1.3}{$y\!=\! 50 \Big(\mathrm{fit2}(x)-\mathrm{maga}(x)\Big)$}}
  +
\end{picture}\end{document}
  +
</nowiki></nomathjax></poem>
  +
  +
==References==
  +
<references/>
  +
  +
[[Category:Bessel transform]]
  +
[[Category:C++]]
  +
[[Category:Explicit plot]]
  +
[[Category:FFT]]
  +
[[Category:Fragment]]
  +
[[Category:Latex]]
  +
[[Category:Maga function]]
  +
[[Category:Makoto Morinaga]]
  +
[[Category:Mori]]
  +
[[Category:Morinaga function]]
  +
[[Category:Naga]]
  +
[[Category:SinFT]]
  +
[[Category:CosFT]]

Latest revision as of 08:42, 1 December 2018

Fragment of the image http://mizugadro.mydns.jp/t/index.php/File:Magaplot300.jpg

Explicit plot of function maga and related functions.

Thin green curve: $~ y=\,$ nori$(x)$ $\,=\,$ mori$\big(\sqrt{x}\big)^2\,=\,$ $\displaystyle \frac{J_0\big(L\, \sqrt{x}\big)^2}{(1\!-\!x)^2}$

where $J_0=\,$BesselJ0, the zeroth Bessel function, and $L$ is its first zero, $L=\,$BesselJZero$[0,1]$.

The thick blue and thick red lines represent the half of the Bessel transform of square of mori with quadratic phase;

naga$(x) = \displaystyle 2 \,\int_0^\infty \, $mori$(p )^2 \, \exp(\mathrm i \,x \,p^2) \, p \mathrm d p$ $\, =\displaystyle \int_0^\infty \, $nori$(q) \, \exp(\mathrm i \,x \,q)\, \mathrm d q$

The thick black curve shows the maga function,

$y=\,$maga$(x) \, = 1-|$naga$(x)|^2$

The dotted curve shows its fit,

$y=\mathrm{fit2}(x)=\displaystyle \frac{ x^{3/2}}{\sqrt{5-x/2+x^2/4+x^3}}$

The thin blue line shows deviation of this fit from the maga function; in order to mage this deviation visible, it is scaled with factor 50:

$y=50\big( \mathrm{fit2}(x)-\mathrm{maga}(x)\big)$

C++ generator of curves


#include<math.h>
#include<stdio.h>
#include <stdlib.h>
#include <complex>
//using namespace std;
#define z_type std::complex<double>
#define Re(x) x.real()
#define Im(x) x.imag()
#define RI(x) x.real(),x.imag()
#define DB double
#define DO(x,y) for(x=0;x<y;x++)

void fft(z_type *a, int N, int o) // Arry is FIRST!
{z_type u,w,t; int i,j,k,l,e=1,L,p,q,m;
q=N/2; p=2; for(m=1;p<N;m++) p*=2;
p=N-1; z_type y=z_type(0.,M_PI*o); j=0;
for(i=0;i<p;i++){ if(i<j) { t=a[j]; a[j]=a[i]; a[i]=t;}
                   k=q; while(k<=j){ j-=k; k/=2;}
                   j+=k; }
for(l=0;l<m;l++)
{ L=e; e*=2; u=1.; w=exp(y/((double)L));
  for(j=0;j<L;j++)
  { for(i=j;i<N;i+=e){k=i+L; t=a[k]*u; a[k]=a[i]-t; a[i]+=t;}
     u*=w;
} }
}

void ado(FILE *O, int X, int Y)
{ fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
        fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
        fprintf(O,"/M {moveto} bind def\n");
        fprintf(O,"/L {lineto} bind def\n");
        fprintf(O,"/S {stroke} bind def\n");
        fprintf(O,"/s {show newpath} bind def\n");
        fprintf(O,"/C {closepath} bind def\n");
        fprintf(O,"/F {fill} bind def\n");
        fprintf(O,"/o {.01 0 360 arc C S} bind def\n");
        fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
        fprintf(O,"/W {setlinewidth} bind def\n");
        fprintf(O,"/RGB {setrgbcolor} bind def\n");}

// #include "ado.cin"
// #include"fafo.cin"
//#include "mori.cin"
DB L1= 2.404825557695773;
DB L2= 5.5200781102863115;
DB L3= 8.653727912911013;
DB morin(DB x){ return j0(L1*x)/(1-x*x);}
DB mori0(DB x){ int n,m; DB s, xx=x*x;
DB c[16]={ 1., -0.4457964907366961303, 0.07678538241994023453, -0.0071642885058902232688,
0.00042159522055140947688, -0.000017110542281627483109, 5.0832583976057607495e-7, -1.1537378620148452816e-8,
2.0662789231930073316e-10, -2.9948657413756059965e-12, 3.5852738451127332173e-14,-3.6050239634659700777e-16,
3.0877184831292878827e-18, -2.2798156440952688462e-20, 1.4660907878585489441e-22,-8.2852774398657968065e-25};
// 16th term seems to fail; perhaps, due to the C++ rounding errors.
//with m=15, at |x|<2, the error is of order of 10^(-16)
//In this sense, the result is accurate while |x|<2.
m=15; s=c[m]*xx; for(n=m-1;n>0;n--){ s+=c[n]; s*=xx;}
return 1.+s;}
DB mori(DB x){if(fabs(x)<2) return mori0(x);
                        return morin(x);}

DB fit(DB x){ DB a,b,c,cc,s; a=-.055; b=0.02; c=0.45; cc=c*c;
s=x*(a+x*(b+x*cc)); return c*x*sqrt(x/(1.+s)); }

DB fit2(DB x){ DB q; q=5.+x*(-.5+x*(.25+x)); return x*sqrt(x/q);}

int main(){z_type * a, *b, c; int j,m,n,N; FILE *o; DB scale,step,x,y,q,p,u,v;
n=15;
N=pow(2,n);
//scale=.5;
step=sqrt(2*M_PI/N);
scale=100*step;
DB dx=step*scale;
DB dp=step/scale;
printf("2^%2d=%8d scale=%6.4lf dx=%9.8lf dp=%9.8lf\n",n,N,scale,dx,dp);
a=(z_type *) malloc((size_t)((N+1)*sizeof(z_type)));
b=(z_type *) malloc((size_t)((N+1)*sizeof(z_type)));

DO(m,N){x=m*dx; y=mori(sqrt(x)); y*=y; a[m]=y; b[m]=y*dx; }
b[0]*=.5;
fft(b,N,1); //DO(j,N) printf("%2d %18.15f %18.15f %18.15f %18.15f\n", j, RI(a[j]), RI(b[j]) );
//o=fopen("32.eps","w"); ado(o,1008,228);
o=fopen("magaplo.eps","w"); ado(o,1008,208);
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
#define o(x,y) fprintf(o,"%6.4f %6.4f o\n",0.+x,0.+y);
fprintf(o,"4 104 translate 100 100 scale\n 1 setlinejoin 2 setlinecap\n");
for(m=0;m<11;m++) {M(m,-1) L(m,1)}
for(n=-1;n<2;n++) {M(0,n) L(10,n)} fprintf(o,".002 W S\n");

DO(m,N){x=dx*m; y=Re(a[m]); if(m==0)M(x,y)else L(x,y);if(x>10) break;} fprintf(o,".008 W 0 .8 0 RGB S\n");
//DO(n,N){p=dp*n; y=fit(p); if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".008 W 1 0 0 RGB S\n");
//DO(n,N){p=dp*n; y=fit2(p); if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".008 W 0 0 1 RGB S\n");
fprintf(o,".01 W 0 0 1 RGB\n");
for(n=4;n<N;n+=4){p=dp*n; y=fit2(p); o(p,y);if(p>10) break;}

DO(n,N){p=dp*n; y=Re(b[n]); if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".02 W 0 0 1 RGB S\n");
DO(n,N){p=dp*n; y=Im(b[n]); if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".02 W 1 0 0 RGB S\n");
DO(n,N){p=dp*n; u=Re(b[n]); v=Im(b[n]); y=1-u*u-v*v; if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".02 W 0 0 0 RGB S\n");
// fprintf(o,".01 W 0 0 0 RGB\n");
//for(n=4;n<N;n+=4){p=dp*n; u=Re(b[n]); v=Im(b[n]); y=1-u*u-v*v; o(p,y);if(p>10) break;}

//DO(n,N){p=dp*n; u=Re(b[n]); v=Im(b[n]); y=fit(p)-(1.-u*u-v*v); y*=50; if(n==0)M(p,y)else L(p,y);if(p>10) break;} fprintf(o,".004 W .8 0 0 RGB S\n");
DO(n,N){p=dp*n; u=Re(b[n]); v=Im(b[n]); y=fit2(p)-(1.-u*u-v*v); y*=50; if(n==0)M(p,y)else L(p,y);if(p>10) break;}
fprintf(o,".008 W 0 0 1 RGB S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
 system("epstopdf magaplo.eps");
 system( "open magaplo.pdf");
 free(a);
 free(b);
}

Latex generator of labels


\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphics}
\usepackage{rotating}
\paperwidth 420pt
\paperheight 216pt
\textwidth 1420pt
\textheight 300pt
\topmargin -108pt
\oddsidemargin -73pt
\newcommand \ds {\displaystyle}
\newcommand \sx {\scalebox}
\newcommand \rme {\mathrm{e}}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\pagestyle{empty}
\parindent 0pt
\begin{document}
\begin{picture}(410,214)
\put(10,0){\ing{magaplo}}
%\put(.6,120){\sx{1.25}{$y$}}
\put(2,199){\sx{1.3}{$1$}}
%\put(.6,56){\sx{1.4}{$\frac{1}{2}$}}
\put(2, 100){\sx{1.3}{$0$}}
\put(-2, 0){\sx{1.3}{$-\!1$}}
\put(111,88){\sx{1.33}{$1$}}
\put(211,88){\sx{1.33}{$2$}}
\put(311,89){\sx{1.33}{$3$}}
\put(406,89){\sx{1.4}{$x$}}
%\put(411,89){\sx{1.33}{$4$}}
%\put(512,89){\sx{1.33}{$5$}}
%\put(613,89){\sx{1.33}{$6$}}
%\put(713,89){\sx{1.33}{$7$}}
%\put(813,89){\sx{1.33}{$8$}}
%\put(914,89){\sx{1.33}{$9$}}
%\put(1010,89){\sx{1.4}{$x$}}
%\put(15,192){\rot{-36}\sx{1.3}{$y\!=\! \mathrm{nori}(x)$}\ero}

\put(280,199){\rot{3}\sx{1.3}{$y\!=\! \mathrm{fit2}(x))$}\ero}
\put(280,178){\rot{3}\sx{1.3}{$y\!=\! \mathrm{maga}(x))$}\ero}

\put(118,130){\rot{-16}\sx{1.3}{$y\!=\! \mathrm{nori}(x)$}\ero}
\put(292,142){\rot{-4}\sx{1.3}{$y\!=\! \Im(\mathrm{naga}(x))$}\ero}
\put(292,120){\rot{-4}\sx{1.3}{$y\!=\! \Re(\mathrm{naga}(x))$}\ero}

\put(170,26){\sx{1.3}{$y\!=\! 50 \Big(\mathrm{fit2}(x)-\mathrm{maga}(x)\Big)$}}
\end{picture}\end{document}

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