fn[q_]=(( 2Gamma[2 q-1])/(4^q Gamma[q]^2));

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2))),{m,-∞,+∞}]]

Limit[(1/R)r^1tmp/.{r->(x/R)},x->∞]

fn[1]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)^2)),{m,-∞,+∞}]]

Limit[(1/R)r^3tmp/.{r->(x/R)},x->∞]

fn[2]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)^3)),{m,-∞,+∞}]]

Limit[(1/R)r^5tmp/.{r->(x/R)},x->∞]

fn[3]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)^4)),{m,-∞,+∞}]]

Limit[(1/R)r^7tmp/.{r->(x/R)},x->∞]

fn[4]

tmp=Integrate[σ^((5-p)/2)Exp[-I σ r^2],{σ,0,∞}]

((Gamma[(7/2)-(p/2)])/((4 π)^((p+1)/2)))/.{p->0}

6 fn[(8/2)]

((Gamma[(7/2)-(p/2)])/((4 π)^((p+1)/2)))/.{p->1}

6 fn[(8/2)]fn[(7/2)]

((Gamma[(7/2)-(p/2)])/((4 π)^((p+1)/2)))/.{p->2}

6fn[(8/2)]fn[(7/2)]fn[(6/2)]

((Gamma[(7/2)-(p/2)])/( (4 π)^((p+1)/2)))/.{p->3}

6fn[(8/2)]fn[(7/2)]fn[(6/2)]fn[(5/2)]

fn[(8/2)]

fn[(8/2)]

(1/6)((Gamma[(7/2)-(p/2)])/((4 π)^((p+1)/2)))/.{p->0}

fn[(8/2)]fn[(7/2)]

(1/6)((Gamma[(7/2)-(p/2)])/((4 π)^((p+1)/2)))/.{p->1}

fn[(8/2)]fn[(7/2)]fn[(6/2)]

(1/6)((Gamma[(7/2)-(p/2)])/((4 π)^((p+1)/2)))/.{p->2}

fn[(8/2)]fn[(7/2)]fn[(6/2)]fn[(5/2)]

(1/6)((Gamma[(7/2)-(p/2)])/((4 π)^((p+1)/2)))/.{p->3}

Series[Log[1+x],{x,0,4}]

Integrate[Exp[-(1/(4 τ))x^2],{x,-∞,∞}]

Integrate[Exp[-(1/(4 τ))x^2],{x,-∞,∞}]

Expand[TrigToExp[16Sin[2 x]^4]]

((Gamma[7-p])/((4π)^(p/2)2^(7-p)Gamma[((7-p)/2)]))/.{p->0}

(1/((4π)^((1+p)/2)))Gamma[((7-p)/2)]/.{p->0}

Integrate[σ^(((6-p)/2)) Exp[-σ μ^2],{σ,0,∞}]

Gamma[4]

tmp=Sum[(1/(((((2π n)/R))^2+μ^2)^(4-p/2)))/.{p->0},{n,-∞,∞}];

Limit[μ^7tmp/.{μ->(x/R)},x->∞]

(2/(4^(4-p/2)))((Gamma[2(4-p/2)-1])/(Gamma[4-p/2]^2))/.{p->0}

2Gamma[]

Gamma[4]*((5 R)/32)

Gamma[4-p/2](2/(4^(4-p/2)))((Gamma[2(4-p/2)-1])/(Gamma[4-p/2]^2))/.{p->0}

(1/(2^(7-p)))((Gamma[7-p])/(Gamma[4-p/2]))/.{p->0}

Table[{p,((Gamma[7-p])/(Gamma[4-p/2])),((2^(7-p))/(Sqrt[4π]))Gamma[((7-p)/2)]},{p,0,6}]

FullSimplify[6+Exp[2 x]+Exp[-2x]-4Exp[x]-4Exp[-x]]

FullSimplify[6+Exp[2 x]+Exp[-2x]]

FullSimplify[-4Exp[x]-4Exp[-x]]

Simplify[Series[Sqrt[((2v)/(Sinh[2 v σ]))],{v,0,4}]]

Simplify[Series[2 (3+Cosh[2 v σ]),{v,0,4}]]

Simplify[Series[v Tanh[v σ],{v,0,4}]]

FullSimplify[Series[Sqrt[((2v)/(Sinh[2 v σ]))]2 (3+Cosh[2 v σ])Exp[-v Tanh[v σ]x^2],{v,0,4}]]

FullSimplify[Series[Sqrt[((2v)/(Sinh[2 v σ]))](-8 Cosh[v σ])Exp[-v Tanh[v σ]x^2],{v,0,4}]]

Simplify[Series[v Tanh[σ v],{v,0,4}]]

Integrate[Exp[-v Tanh[σ v]x^2],{x,-∞,+∞}]

Simplify[Series[((Sqrt[π])/(Sqrt[v Tanh[v σ]])),{v,0,4}]]

FullSimplify[Sum[Exp[-σ(m^2+μ^2)],{m,-∞,+∞}]]

Simplify[Series[Exp[-v Tanh[v σ]x^2],{v,0,4}]]

Integrate[σ^(-(1+p)/2)((8/(Sqrt[σ])))Exp[-σ μ^2],{σ,0,∞}]

Integrate[σ^(-(1+p)/2)((4/3)(σ-6x^2)Sqrt[σ])Exp[-σ μ^2],{σ,0,∞}]

Integrate[σ^(-(1+p)/2)((4/15)(15x^4+5x^2σ+3σ^2)σ^(3/2))Exp[-σ μ^2],{σ,0,∞}]

Integrate[σ^(-(1+p)/2)((1/15)(-60x^4-20x^2σ+3σ^2)σ^(3/2))Exp[-σ μ^2],{σ,0,∞}]

Bosons

fn[q_]=(( 2Gamma[2 q-1])/(4^q Gamma[q]^2));

fn2[q_]=(1/(2^(q-1) Gamma[q]));

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2))),{m,-∞,+∞}]]

Limit[(1/R)r^1tmp/.{r->(x/R)},x->∞]

fn[1]

Limit[Exp[r R]((1/R)r^1tmp-fn[1])/.{r->(x/R)},x->∞]

fn2[1]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)^2)),{m,-∞,+∞}]]

Limit[(1/R)r^3tmp/.{r->(x/R)},x->∞]

fn[2]

Limit[(1/(r R)) Exp[r R]((1/R)r^3tmp-fn[2])/.{r->(x/R)},x->∞]

fn2[2]

TrigToExp[Sinh[x]]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)^3)),{m,-∞,+∞}]]

Limit[(1/R)r^5tmp/.{r->(x/R)},x->∞]

fn[3]

Limit[(1/((r R)^2)) Exp[r R]((1/R)r^5tmp-fn[3])/.{r->(x/R)},x->∞]

fn2[3]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)^4)),{m,-∞,+∞}]]

Limit[(1/R)r^7tmp/.{r->(x/R)},x->∞]

fn[4]

Limit[(1/((r R)^3)) Exp[r R]((1/R)r^7tmp-fn[4])/.{r->(x/R)},x->∞]

fn2[4]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)^5)),{m,-∞,+∞}]]

Limit[(1/R)r^9tmp/.{r->(x/R)},x->∞]

fn[5]

Limit[(1/((r R)^4)) Exp[r R]((1/R)r^9tmp-fn[5])/.{r->(x/R)},x->∞]

fn2[5]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)^6)),{m,-∞,+∞}]]

Limit[(1/R)r^11tmp/.{r->(x/R)},x->∞]

fn[6]

Limit[(1/((r R)^5)) Exp[r R]((1/R)r^11tmp-fn[6])/.{r->(x/R)},x->∞]

fn2[6]

Fermions

tmp=Simplify[Sum[(1/(((((2π m)/R)+(π/R))^2+r^2))),{m,-∞,+∞}]]

Limit[(1/R)r^1tmp/.{r->(x/R)},x->∞]

fn[1]

Limit[Exp[r R]((1/R)r^1tmp-fn[1])/.{r->(x/R)},x->∞]

tmp=Simplify[Sum[(1/(((((2π m)/R)+(π/R))^2+r^2)^2)),{m,-∞,+∞}]]

Limit[(1/R)r^3tmp/.{r->(x/R)},x->∞]

fn[2]

Limit[(1/(r R)) Exp[r R]((1/R)r^3tmp-fn[2])/.{r->(x/R)},x->∞]

tmp=Simplify[Sum[(1/(((((2π m)/R)+(π/R))^2+r^2)^3)),{m,-∞,+∞}]]

Limit[(1/R)r^5tmp/.{r->(x/R)},x->∞]

fn[3]

Limit[(1/((r R)^2)) Exp[r R]((1/R)r^5tmp-fn[3])/.{r->(x/R)},x->∞]

tmp=Simplify[Sum[(1/(((((2π m)/R)+(π/R))^2+r^2)^4)),{m,-∞,+∞}]]

Limit[(1/R)r^7tmp/.{r->(x/R)},x->∞]

fn[4]

Limit[(1/((r R)^3)) Exp[r R]((1/R)r^7tmp-fn[4])/.{r->(x/R)},x->∞]

Bosons-Fermions

fn3[q_]=(1/(2^(q-2) Gamma[q]));

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)))-(1/(((((2π m)/R)+(π/R))^2+r^2))),{m,-∞,+∞}]]

Limit[((r/R))^1Exp[r R]tmp/.{r->(x/R)},x->∞]

fn3[1]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)))-(1/(((((2π m)/R)+(π/R))^2+r^2))),{m,-∞,+∞}]]

Limit[((r/R))^1Exp[r R]tmp/.{r->(x/R)},x->∞]

fn3[1]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)^2))-(1/(((((2π m)/R)+(π/R))^2+r^2)^2)),{m,-∞,+∞}]]

Limit[((r/R))^2Exp[r R]tmp/.{r->(x/R)},x->∞]

fn3[2]

tmp=Simplify[Sum[(1/(((((2π m)/R))^2+r^2)^5))-(1/(((((2π m)/R)+(π/R))^2+r^2)^5)),{m,-∞,+∞}]]

Limit[((r/R))^5Exp[r R]tmp/.{r->(x/R)},x->∞]

fn3[5]