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I extract the function jin[r] by solving eqsynin, and then I develop the function's series (around zero) to generate an equation for m1in and m2in based on esyn and gamma, knowing that the function jin[r] has a value of 1 for r=0. Basically, after applying various initial conditions, I extract the values esynm1in and m2in and graph the jin[r] for different gamma values to verify the theory of the function's behaviour for large and small gamma values.

In the beginning without defining the gamma the expression I get in the function and Series is very complicated and I cant derive an expression for m1in and m2in based on esyn and gamma.

FullSimplify[DSolve[eqsynin == 0, jin[r], r]] /. {C[1] -> m1in, 
  C[2] -> m2in}
{{jin[r] -> (1/(gamma^(3/2) r^1. (-gamma r^2)^4.))
   E^(-1.24976 Sqrt[gamma]
      r) ((-gamma r^2)^4. (1. gamma^(3/2)
           m1in + (-0.226362 esyn - 0.173716 gamma) Gamma[
           2., -1.24976 Sqrt[gamma] r] + 
         E^(2.49951 Sqrt[gamma]
            r) (0.400078 gamma m2in + (0.226362 esyn + 
               0.173716 gamma) Gamma[2., 1.24976 Sqrt[gamma] r])) + 
      gamma r^4. (0.144928 (Sqrt[gamma] r)^4. Gamma[
           4., -1.24976 Sqrt[gamma] r] - 
         0.144928 E^(
          2.49951 Sqrt[gamma] r) (-Sqrt[gamma] r)^4. Gamma[4., 
           1.24976 Sqrt[gamma] r]))}}
jin[r_] := (E^(-1.2497571136095054` Sqrt[gamma]
      r) ((-gamma r^2)^4.` (1.` gamma^(3/2)
           m1in + (-0.2263621298822344` esyn - 
            0.17371560886807141` gamma) GammaRegularized[
           2.`, -1.2497571136095054` Sqrt[gamma] r] + 
         E^(2.4995142272190107` Sqrt[gamma]
            r) (0.40007773875030583` gamma m2in + \
(0.22636212988223445` esyn + 
               0.17371560886807141` gamma) GammaRegularized[2.`, 
              1.2497571136095054` Sqrt[gamma] r])) + 
      gamma r^4.` (0.14492807934563867` (Sqrt[gamma] r)^4.` Gamma[
           4.`, -1.2497571136095054` Sqrt[gamma] r] - 
         0.14492807934563867` E^(
          2.4995142272190107` Sqrt[gamma]
            r) (-Sqrt[gamma] r)^4.` GammaRegularized[4.`, 
           1.2497571136095054` Sqrt[gamma] r])))/(gamma^(3/2)
     r^1.` (-gamma r^2)^4.`)

Series[jin[r], {r, 0, 0}]
1/(
 r^1. (-gamma r^2)^4.) (E^((0. + 12.5664 I) Floor[
      0.159155 (3.14159 - 1. Arg[-Sqrt[gamma]] - 
         1. Arg[r])]) (-gamma r^2)^4. SeriesData[r, 0, {}, 2, 2, 1] + 
   E^((0. + 12.5664 I) Floor[
      0.159155 (3.14159 - 0.5 Arg[gamma] - 
         1. Arg[r])]) (-gamma r^2)^4. SeriesData[r, 0, {}, 2, 2, 1] + 
   E^((0. + 25.1327 I) Floor[
      0.159155 (3.14159 - 0.5 Arg[gamma] - 1. Arg[r])])
     r^4. (-Sqrt[gamma] r)^4. SeriesData[r, 0, {}, 3, 3, 1] + 
   E^((0. + 25.1327 I) Floor[
      0.159155 (3.14159 - 1. Arg[-Sqrt[gamma]] - 1. Arg[r])])
     r^4. (Sqrt[gamma] r)^4. SeriesData[r, 0, {}, 4, 4, 1] + 
   r^4. (-Sqrt[gamma] r)^4. (
SeriesData[
     r, 0, {(-0.14492807934563867`) gamma^Rational[-1, 2]}, 0, 1, 
      1]) + r^4. (Sqrt[gamma] r)^4. (
SeriesData[
     r, 0, {0.869568476073832 gamma^Rational[-1, 2]}, 0, 1, 
      1]) + (-gamma r^2)^4. (
SeriesData[
     r, 0, {1. gamma^Rational[-3, 2] (
        5.551115123125783*^-17 esyn + 1. gamma^Rational[3, 2]
            m1in + 0.40007773875030583` gamma m2in)}, 0, 1, 1]))

Then I defined the gamma and I observed that for some values of gamma the function jin[r] and Serie's expression was more simplified and I could then move forward by expressing m1in and m2in based on esyn.

Clear[gamma, jin]

gamma := 25

FullSimplify[DSolve[eqsynin == 0, jin[r], r]] /. {C[1] -> m1in, 
  C[2] -> m2in}
{{jin[r] -> 
   0.430727 + esyn (0.0226318 + 4.33681*10^-19/r) + (
    E^(-6.24879 r) (m1in + E^(12.4976 r) m2in))/r + 
    1.73472*10^-18 r - 0.0226318 r^2}}
jin[r_] := 
 0.43072703781177735` + 
  esyn (0.022631814565769705` + 4.336808689942018`*^-19/r) + (
  E^(-6.248785568047525` r) (m1in + E^(12.497571136095052` r) m2in))/
  r + 1.734723475976807`*^-18 r - 0.02263181456576971` r^2

Series[jin[r], {r, 0, 0}]
SeriesData[r, 0, \
{4.336808689942018*^-19 esyn + m1in + m2in, 
  0.43072703781177735` + 0.022631814565769705` esyn + 12.\
497571136095052` m2in - 6.248785568047525 (m1in + m2in)}, -1, 1, 1]
Clear[gamma, jin]

gamma := 24

FullSimplify[DSolve[eqsynin == 0, jin[r], r]] /. {C[1] -> m1in, 
  C[2] -> m2in}
{{jin[r] -> 
   1/(-r)^5. E^(-6.12253 r) (-0.0000513599 r^4. Gamma[
        4., -6.12253 r] + (-r)^4. (-1. m1in + (0.0354596 + 
            0.00192525 esyn) Gamma[2., -6.12253 r] + 
         E^(12.2451 r) (-0.0816655 m2in + (-0.0354596 - 
               0.00192525 esyn) Gamma[2., 6.12253 r] + 
            0.0000513599 Gamma[4., 6.12253 r])))}}
jin[r_] := 
 1/(-r)^5.` E^(-6.122534461513589` r) (-0.00005135993110509381` r^4.` \
Gamma[4.`, -6.122534461513589` r] + (-r)^4.` (-1.` m1in + \
(0.03545955017363964` + 0.0019252490114621068` esyn) Gamma[
         2.`, -6.122534461513589` r] + 
       E^(12.245068923027178` r) (-0.08166552644873018` m2in + \
(-0.03545955017363964` - 0.001925249011462107` esyn) Gamma[2.`, 
            6.122534461513589` r] + 
          0.00005135993110509382` Gamma[4.`, 6.122534461513589` r])))

Series[jin[r], {r, 0, 0}]
(
 E^((0. + 12.5664 I) Floor[-0.159155 Arg[r]]) SeriesData[
  r, 0, {}, 2, 2, 1])/(-r)^1. + (
 E^((0. + 25.1327 I) Floor[-0.159155 Arg[r]]) r^4. SeriesData[
  r, 0, {}, 4, 4, 1])/(-r)^5. + (r^4. (
SeriesData[r, 0, {
Complex[-0.018042195912175808`, 0.]}, 0, 1, 1]))/(-r)^1. + (r^4. (
SeriesData[r, 0, {-0.0003081595866305629}, 0, 1, 1]))/(-r)^5. + (r^2. (
SeriesData[r, 0, {
Complex[0.6646082115939588, 0.]}, 0, 1, 1]))/(-r)^1. + (r^2. (
SeriesData[
   r, 0, {0.036084391824351615` esyn}, 0, 1, 
    1]))/(-r)^1. + SeriesData[
 r, 0, {0.00030815958663056586` - 2.168404344971009*^-19 esyn - 1.\
 m1in - 0.08166552644873018 m2in}, 0, 1, 1]/(-r)^1.
Clear[gamma, jin]

gamma := 23

FullSimplify[DSolve[eqsynin == 0, jin[r], r]] /. {C[1] -> m1in, 
  C[2] -> m2in}
{{jin[r] -> 
   0.430096 + 0.0245998 esyn + (
    6.93889*10^-18 + E^(-5.99362 r) m1in + E^(5.99362 r) m2in)/r - 
    1.73472*10^-18 r - 0.0245998 r^2}}
jin[r_] := 
 0.43009594251457106` + 0.02459979844105403` esyn + (
  6.938893903907228`*^-18 + E^(-5.993624561932782` r) m1in + 
   E^(5.99362456193278` r) m2in)/r - 1.734723475976807`*^-18 r - 
  0.02459979844105403` r^2

Series[jin[r], {r, 0, 0}]
SeriesData[r, 0, {6.938893903907228*^-18 + m1in + m2in, 
  0.43009594251457106` + 0.02459979844105403 esyn - 5.\
993624561932782 m1in + 5.99362456193278 m2in}, -1, 1, 1]
Clear[gamma, jin]

gamma := 22

FullSimplify[DSolve[eqsynin == 0, jin[r], r]] /. {C[1] -> m1in, 
  C[2] -> m2in}
{{jin[r] -> 
   1/(-r)^5. E^(-5.86188 r) (-0.0000638404 r^4. Gamma[
        4., -5.86188 r] + (-r)^4. (-1. m1in + (0.0370363 + 
            0.00219366 esyn) Gamma[2., -5.86188 r] + 
         E^(11.7238 r) (-0.0852969 m2in + (-0.0370363 - 
               0.00219366 esyn) Gamma[2., 5.86188 r] + 
            0.0000638404 Gamma[4., 5.86188 r])))}}
jin[r_] := 
 1/(-r)^5.` E^(-5.861880461625464` r) (-0.00006384043457961346` r^4.` \
Gamma[4.`, -5.861880461625464` r] + (-r)^4.` (-1.` m1in + \
(0.03703629225282387` + 0.0021936621930306417` esyn) Gamma[
         2.`, -5.861880461625464` r] + 
       E^(11.723760923250929` r) (-0.08529686049949797` m2in + \
(-0.037036292252823864` - 0.0021936621930306413` esyn) Gamma[2.`, 
            5.861880461625464` r] + 
          0.00006384043457961345` Gamma[4.`, 5.861880461625464` r])))

Series[jin[r], {r, 0, 0}]
(
  E^((0. + 12.5664 I) Floor[-0.159155 Arg[r]]) SeriesData[
  r, 0, {}, 2, 2, 1])/(-r)^1. + (
 E^((0. + 25.1327 I) Floor[-0.159155 Arg[r]]) r^4. SeriesData[
  r, 0, {}, 4, 4, 1])/(-r)^5. + (r^4. (
 SeriesData[r, 0, {
Complex[-0.018844459036110223`, 0.]}, 0, 1, 1]))/(-r)^1. + (r^4. (
SeriesData[r, 0, {-0.0003830426074776808}, 0, 1, 1]))/(-r)^5. + (r^2. (
SeriesData[r, 0, {
Complex[0.6363139178175161, 0.]}, 0, 1, 1]))/(-r)^1. + (r^2. (
SeriesData[
   r, 0, {0.03768891807222045 esyn}, 0, 1, 1]))/(-r)^1. + SeriesData[
 r, 0, {0.0003830426074776863 + 4.336808689942018*^-19 esyn - 1.\
 m1in - 0.08529686049949797 m2in}, 0, 1, 1]/(-r)^1.
Clear[gamma, jin]

gamma := 21

FullSimplify[DSolve[eqsynin == 0, jin[r], r]] /. {C[1] -> m1in, 
  C[2] -> m2in}
{{jin[r] -> 
   1/(-r)^5. E^(-5.72711 r) (-0.0000717141 r^4. Gamma[
        4., -5.72711 r] + (-r)^4. (-1. m1in + (0.0379079 + 
            0.0023522 esyn) Gamma[2., -5.72711 r] + 
         E^(11.4542 r) (-0.0873041 m2in + (-0.0379079 - 
               0.0023522 esyn) Gamma[2., 5.72711 r] + 
            0.0000717141 Gamma[4., 5.72711 r])))}}
jin[r_] := 
 1/(-r)^5.` E^(-5.7271065734250834` r) (-0.0000717140583010426` r^4.` \
Gamma[4.`, -5.7271065734250834` r] + (-r)^4.` (-1.` m1in + \
(0.0379078536682514` + 0.0023522031624869945` esyn) Gamma[
         2.`, -5.7271065734250834` r] + 
       E^(11.454213146850167` r) (-0.08730412008047829` m2in + \
(-0.0379078536682514` - 0.0023522031624869945` esyn) Gamma[2.`, 
            5.7271065734250834` r] + 
          0.00007171405830104261` Gamma[4.`, 5.7271065734250834` r])))

Series[jin[r], {r, 0, 0}]
(
 E^((0. + 12.5664 I) Floor[-0.159155 Arg[r]]) SeriesData[
  r, 0, {}, 2, 2, 1])/(-r)^1. + (
 E^((0. + 25.1327 I) Floor[-0.159155 Arg[r]]) r^4. SeriesData[
  r, 0, {}, 4, 4, 1])/(-r)^5. + (r^4. (
SeriesData[r, 0, {
Complex[-0.019287918745261493`, 0.]}, 0, 1, 1]))/(-r)^1. + (r^4. (
SeriesData[r, 0, {-0.00043028434980625554`}, 0, 1, 1]))/(-r)^5. + (
 r^2. (
SeriesData[r, 0, {
Complex[0.6216840560552884, 0.]}, 0, 1, 1]))/(-r)^1. + (r^2. (
SeriesData[
   r, 0, {0.03857583749052298 esyn}, 0, 1, 1]))/(-r)^1. + SeriesData[
 r, 0, {0.0004302843498062564 - 1. m1in - 0.08730412008047829 m2in}, 
  0, 1, 1]/(-r)^1.

I need your assistance to help me derive an expression for m1in and m2in based on esyn and gamma from the generic Series of jin[r] (without defining the value of gamma) and explain me why for different values of gamma I get more simplify equation of jin[r]. I think that the 'problem' has to do with the gamma function.

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