Editing Gamma function (section)

=== Minima and maxima ===
On the real line, the gamma function has a local minimum at {{math|''z''<sub>min</sub> ≈ {{gaps|+1.46163|21449|68362|34126}}}}<ref>{{Cite OEIS|A030169|2=Decimal expansion of real number x such that y = Gamma(x) is a minimum}}</ref> where it attains the value {{math|Γ(''z''<sub>min</sub>) ≈ {{gaps|+0.88560|31944|10888|70027}}}}.<ref>{{Cite OEIS|A030171|2=Decimal expansion of real number y such that y = Gamma(x) is a minimum}}</ref> The gamma function rises to either side of this minimum. The solution to {{math|1=Γ(''z'' − 0.5) = Γ(''z'' + 0.5)}} is {{math|1=''z'' = +1.5}} and the common value is {{math|1=Γ(1) = Γ(2) = +1}}.  The positive solution to {{math|1=Γ(''z'' − 1) = Γ(''z'' + 1)}} is {{math|1=''z'' = ''φ'' ≈ +1.618}}, the [[golden ratio]], and the common value is {{math|1=Γ(''φ'' − 1) = Γ(''φ'' + 1) = ''φ''! ≈ {{gaps|+1.44922|96022|69896|60037}}}}.<ref>{{Cite OEIS|A178840|Decimal expansion of the factorial of Golden Ratio}}</ref>

The gamma function must alternate sign between its poles at the non-positive integers because the product in the forward recurrence contains an odd number of negative factors if the number of poles between <math>z</math> and <math>z + n</math> is odd, and an even number if the number of poles is even.<ref name="Mathworld" />  The values at the local extrema of the gamma function along the real axis between the non-positive integers are:
: {{math|1=Γ({{gaps|−0.50408|30082|64455|40925...}}<ref>{{Cite OEIS|A175472|Decimal expansion of the absolute value of the abscissa of the local maximum of the Gamma function in the interval [ -1,0]}}</ref>) = {{gaps|−3.54464|36111|55005|08912...}}}},
: {{math|1=Γ({{gaps|−1.57349|84731|62390|45877...}}<ref>{{Cite OEIS|A175473|Decimal expansion of the absolute value of the abscissa of the local minimum of the Gamma function in the interval [ -2,-1]}}</ref>) =  {{gaps|2.30240|72583|39680|13582...}}}},
: {{math|1=Γ({{gaps|−2.61072|08684|44144|65000...}}<ref>{{Cite OEIS|A175474|Decimal expansion of the absolute value of the abscissa of the local maximum of the Gamma function in the interval [ -3,-2]}}</ref>) = {{gaps|−0.88813|63584|01241|92009...}}}},
: {{math|1=Γ({{gaps|−3.63529|33664|36901|09783...}}<ref>{{Cite OEIS|A256681|Decimal expansion of the [negated] abscissa of the Gamma function local minimum in the interval [-4,-3]}}</ref>) =  {{gaps|0.24512|75398|34366|25043...}}}},
: {{math|1=Γ({{gaps|−4.65323|77617|43142|44171...}}<ref>{{Cite OEIS|A256682|Decimal expansion of the [negated] abscissa of the Gamma function local maximum in the interval [-5,-4]}}</ref>) = {{gaps|−0.05277|96395|87319|40076...}}}}, etc.