# Weird expression for function Series-Expansion with Gamma function for different values of gamma coefficient

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.