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Particular values of the gamma function
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<h2 id="integers-and-half-integers">Integers and half-integers</h2> <p>For positive integer arguments, the gamma function coincides with the <a href="/facts/Factorial/kkfy1Bwe">factorial</a>. That is, </p> Γ ( n ) = ( n − 1 ) ! , {\displaystyle \Gamma (n)=(n-1)!,} <p>and hence </p> Γ ( 1 ) = 1 , Γ ( 2 ) = 1 , Γ ( 3 ) = 2 , Γ ( 4 ) = 6 , Γ ( 5 ) = 24 , {\displaystyle {\begin{aligned}\Gamma (1)&=1,\\\Gamma (2)&=1,\\\Gamma (3)&=2,\\\Gamma (4)&=6,\\\Gamma (5)&=24,\end{aligned}}} <p>and so on. For non-positive integers, the gamma function is not defined. </p><p>For positive half-integers k 2 {\displaystyle {\frac {k}{2}}} where k ∈ 2 N ∗ + 1 {\displaystyle k\in 2\mathbb {N} ^{*}+1} is an odd integer greater or equal 3 {\displaystyle 3} , the function values are given exactly by </p> Γ ( k 2 ) = π ( k − 2 ) ! ! 2 k − 1 2 , {\displaystyle \Gamma \left({\tfrac {k}{2}}\right)={\sqrt {\pi }}{\frac {(k-2)!!}{2^{\frac {k-1}{2}}}}\,,} <p>or equivalently, for non-negative integer values of n: </p> Γ ( 1 2 + n ) = ( 2 n − 1 ) ! ! 2 n π = ( 2 n ) ! 4 n n ! π Γ ( 1 2 − n ) = ( − 2 ) n ( 2 n − 1 ) ! ! π = ( − 4 ) n n ! ( 2 n ) ! π {\displaystyle {\begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={\frac {(2n-1)!!}{2^{n}}}\,{\sqrt {\pi }}={\frac {(2n)!}{4^{n}n!}}{\sqrt {\pi }}\\\Gamma \left({\tfrac {1}{2}}-n\right)&={\frac {(-2)^{n}}{(2n-1)!!}}\,{\sqrt {\pi }}={\frac {(-4)^{n}n!}{(2n)!}}{\sqrt {\pi }}\end{aligned}}} <p>where <i>n</i>!! denotes the <a href="/facts/Double_factorial/VSU8AgzB">double factorial</a>. In particular, </p> <table><tbody><tr><td> Γ ( 1 2 ) {\displaystyle \Gamma \left({\tfrac {1}{2}}\right)\,} </td><td> = π {\displaystyle ={\sqrt {\pi }}\,} </td><td> ≈ 1.772 453 850 905 516 0273 , {\displaystyle \approx 1.772\,453\,850\,905\,516\,0273\,,} </td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A002161</td></tr><tr><td> Γ ( 3 2 ) {\displaystyle \Gamma \left({\tfrac {3}{2}}\right)\,} </td><td> = 1 2 π {\displaystyle ={\tfrac {1}{2}}{\sqrt {\pi }}\,} </td><td> ≈ 0.886 226 925 452 758 0137 , {\displaystyle \approx 0.886\,226\,925\,452\,758\,0137\,,} </td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A019704</td></tr><tr><td> Γ ( 5 2 ) {\displaystyle \Gamma \left({\tfrac {5}{2}}\right)\,} </td><td> = 3 4 π {\displaystyle ={\tfrac {3}{4}}{\sqrt {\pi }}\,} </td><td> ≈ 1.329 340 388 179 137 0205 , {\displaystyle \approx 1.329\,340\,388\,179\,137\,0205\,,} </td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A245884</td></tr><tr><td> Γ ( 7 2 ) {\displaystyle \Gamma \left({\tfrac {7}{2}}\right)\,} </td><td> = 15 8 π {\displaystyle ={\tfrac {15}{8}}{\sqrt {\pi }}\,} </td><td> ≈ 3.323 350 970 447 842 5512 , {\displaystyle \approx 3.323\,350\,970\,447\,842\,5512\,,} </td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A245885</td></tr></tbody></table> <p>and by means of the <a href="/facts/Reflection_formula/aKUhTbH0">reflection formula</a>, </p> <table><tbody><tr><td> Γ ( − 1 2 ) {\displaystyle \Gamma \left(-{\tfrac {1}{2}}\right)\,} </td><td> = − 2 π {\displaystyle =-2{\sqrt {\pi }}\,} </td><td> ≈ − 3.544 907 701 811 032 0546 , {\displaystyle \approx -3.544\,907\,701\,811\,032\,0546\,,} </td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A019707</td></tr><tr><td> Γ ( − 3 2 ) {\displaystyle \Gamma \left(-{\tfrac {3}{2}}\right)\,} </td><td> = 4 3 π {\displaystyle ={\tfrac {4}{3}}{\sqrt {\pi }}\,} </td><td> ≈ 2.363 271 801 207 354 7031 , {\displaystyle \approx 2.363\,271\,801\,207\,354\,7031\,,} </td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A245886</td></tr><tr><td> Γ ( − 5 2 ) {\displaystyle \Gamma \left(-{\tfrac {5}{2}}\right)\,} </td><td> = − 8 15 π {\displaystyle =-{\tfrac {8}{15}}{\sqrt {\pi }}\,} </td><td> ≈ − 0.945 308 720 482 941 8812 , {\displaystyle \approx -0.945\,308\,720\,482\,941\,8812\,,} </td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A245887</td></tr></tbody></table> <h2 id="general-rational-argument">General rational argument</h2> <p>In analogy with the half-integer formula, </p> Γ ( n + 1 3 ) = Γ ( 1 3 ) ( 3 n − 2 ) ! ! ! 3 n Γ ( n + 1 4 ) = Γ ( 1 4 ) ( 4 n − 3 ) ! ! ! ! 4 n Γ ( n + 1 q ) = Γ ( 1 q ) ( q n − ( q − 1 ) ) ! ( q ) q n Γ ( n + p q ) = Γ ( p q ) 1 q n ∏ k = 1 n ( k q + p − q ) {\displaystyle {\begin{aligned}\Gamma \left(n+{\tfrac {1}{3}}\right)&=\Gamma \left({\tfrac {1}{3}}\right){\frac {(3n-2)!!!}{3^{n}}}\\\Gamma \left(n+{\tfrac {1}{4}}\right)&=\Gamma \left({\tfrac {1}{4}}\right){\frac {(4n-3)!!!!}{4^{n}}}\\\Gamma \left(n+{\tfrac {1}{q}}\right)&=\Gamma \left({\tfrac {1}{q}}\right){\frac {{\big (}qn-(q-1){\big )}!^{(q)}}{q^{n}}}\\\Gamma \left(n+{\tfrac {p}{q}}\right)&=\Gamma \left({\tfrac {p}{q}}\right){\frac {1}{q^{n}}}\prod _{k=1}^{n}(kq+p-q)\end{aligned}}} <p>where <i>n</i>!(<i>q</i>) denotes the qth <a href="/facts/Double_factorial/VSU8AgzB">multifactorial</a> of n. Numerically, </p> Γ ( 1 3 ) ≈ 2.678 938 534 707 747 6337 {\displaystyle \Gamma \left({\tfrac {1}{3}}\right)\approx 2.678\,938\,534\,707\,747\,6337} <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A073005 Γ ( 1 4 ) ≈ 3.625 609 908 221 908 3119 {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)\approx 3.625\,609\,908\,221\,908\,3119} <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A068466 Γ ( 1 5 ) ≈ 4.590 843 711 998 803 0532 {\displaystyle \Gamma \left({\tfrac {1}{5}}\right)\approx 4.590\,843\,711\,998\,803\,0532} <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A175380 Γ ( 1 6 ) ≈ 5.566 316 001 780 235 2043 {\displaystyle \Gamma \left({\tfrac {1}{6}}\right)\approx 5.566\,316\,001\,780\,235\,2043} <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A175379 Γ ( 1 7 ) ≈ 6.548 062 940 247 824 4377 {\displaystyle \Gamma \left({\tfrac {1}{7}}\right)\approx 6.548\,062\,940\,247\,824\,4377} <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A220086 Γ ( 1 8 ) ≈ 7.533 941 598 797 611 9047 {\displaystyle \Gamma \left({\tfrac {1}{8}}\right)\approx 7.533\,941\,598\,797\,611\,9047} <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A203142. <p>As n {\displaystyle n} tends to infinity, </p> Γ ( 1 n ) ∼ n − γ {\displaystyle \Gamma \left({\tfrac {1}{n}}\right)\sim n-\gamma } <p>where γ {\displaystyle \gamma } is the <a href="/facts/Euler%25E2%2580%2593Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a> and ∼ {\displaystyle \sim } denotes <a href="/facts/Asymptotic_analysis/kt87qLpu">asymptotic equivalence</a>. </p><p>It is unknown whether these constants are <a href="/facts/Transcendental_number/IrB7JppM">transcendental</a> in general, but Γ(1/3) and Γ(1/4) were shown to be transcendental by <a href="/facts/Chudnovsky_brothers/HXIbzEXL">G. V. Chudnovsky</a>. Γ(1/4) / 4√π has also long been known to be transcendental, and <a href="/facts/Yuri_Valentinovich_Nesterenko/6zyqPEPs">Yuri Nesterenko</a> proved in 1996 that Γ(1/4), π, and <i>e</i>π are <a href="/facts/Algebraically_independent/H8wV21eD">algebraically independent</a>. </p><p>For n ≥ 2 {\displaystyle n\geq 2} at least one of the two numbers Γ ( 1 n ) {\displaystyle \Gamma \left({\tfrac {1}{n}}\right)} and Γ ( 2 n ) {\displaystyle \Gamma \left({\tfrac {2}{n}}\right)} is transcendental.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p><p>The number Γ ( 1 4 ) {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)} is related to the <a href="/facts/Lemniscate_constant/jzkcBdBL">lemniscate constant</a> ϖ {\displaystyle \varpi } by </p> Γ ( 1 4 ) = 2 ϖ 2 π {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)={\sqrt {2\varpi {\sqrt {2\pi }}}}} <p>Borwein and Zucker have found that Γ(<i>n</i>/24) can be expressed algebraically in terms of π, <i>K</i>(<i>k</i>(1)), <i>K</i>(<i>k</i>(2)), <i>K</i>(<i>k</i>(3)), and <i>K</i>(<i>k</i>(6)) where <i>K</i>(<i>k</i>(<i>N</i>)) is a <a href="/facts/Complete_elliptic_integral_of_the_first_kind/q8EGYzQ3">complete elliptic integral of the first kind</a>. This permits efficiently approximating the gamma function of rational arguments to high precision using <a href="/facts/Quadratic_convergence/JVGuzPoS">quadratically convergent</a> <a href="/facts/Arithmetic%25E2%2580%2593geometric_mean/8PRN9UB4">arithmetic–geometric mean</a> iterations. For example: </p> Γ ( 1 6 ) = 3 π Γ ( 1 3 ) 2 2 3 Γ ( 1 4 ) = 2 K ( 1 2 ) π Γ ( 1 3 ) = 2 7 / 9 π K ( 1 4 ( 2 − 3 ) ) 3 3 12 Γ ( 1 8 ) Γ ( 3 8 ) = 8 2 4 ( 2 − 1 ) π K ( 3 − 2 2 ) Γ ( 1 8 ) Γ ( 3 8 ) = 2 ( 1 + 2 ) K ( 1 2 ) π 4 {\displaystyle {\begin{aligned}\Gamma \left({\tfrac {1}{6}}\right)&={\frac {{\sqrt {\frac {3}{\pi }}}\Gamma \left({\frac {1}{3}}\right)^{2}}{\sqrt[{3}]{2}}}\\\Gamma \left({\tfrac {1}{4}}\right)&=2{\sqrt {K\left({\tfrac {1}{2}}\right){\sqrt {\pi }}}}\\\Gamma \left({\tfrac {1}{3}}\right)&={\frac {2^{7/9}{\sqrt[{3}]{\pi K\left({\frac {1}{4}}\left(2-{\sqrt {3}}\right)\right)}}}{\sqrt[{12}]{3}}}\\\Gamma \left({\tfrac {1}{8}}\right)\Gamma \left({\tfrac {3}{8}}\right)&=8{\sqrt[{4}]{2}}{\sqrt {\left({\sqrt {2}}-1\right)\pi }}K\left(3-2{\sqrt {2}}\right)\\{\frac {\Gamma \left({\frac {1}{8}}\right)}{\Gamma \left({\frac {3}{8}}\right)}}&={\frac {2{\sqrt {\left(1+{\sqrt {2}}\right)K\left({\frac {1}{2}}\right)}}}{\sqrt[{4}]{\pi }}}\end{aligned}}} <p>No similar relations are known for Γ(1/5) or other denominators. </p><p>In particular, where AGM() is the <a href="/facts/Arithmetic%25E2%2580%2593geometric_mean/8PRN9UB4">arithmetic–geometric mean</a>, we have<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> Γ ( 1 3 ) = 2 7 9 ⋅ π 2 3 3 1 12 ⋅ AGM ( 2 , 2 + 3 ) 1 3 {\displaystyle \Gamma \left({\tfrac {1}{3}}\right)={\frac {2^{\frac {7}{9}}\cdot \pi ^{\frac {2}{3}}}{3^{\frac {1}{12}}\cdot \operatorname {AGM} \left(2,{\sqrt {2+{\sqrt {3}}}}\right)^{\frac {1}{3}}}}} Γ ( 1 4 ) = ( 2 π ) 3 2 AGM ( 2 , 1 ) {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)={\sqrt {\frac {(2\pi )^{\frac {3}{2}}}{\operatorname {AGM} \left({\sqrt {2}},1\right)}}}} Γ ( 1 6 ) = 2 14 9 ⋅ 3 1 3 ⋅ π 5 6 AGM ( 1 + 3 , 8 ) 2 3 . {\displaystyle \Gamma \left({\tfrac {1}{6}}\right)={\frac {2^{\frac {14}{9}}\cdot 3^{\frac {1}{3}}\cdot \pi ^{\frac {5}{6}}}{\operatorname {AGM} \left(1+{\sqrt {3}},{\sqrt {8}}\right)^{\frac {2}{3}}}}.} <p>Other formulas include the <a href="/facts/Infinite_product/qaaNh5PM">infinite products</a> </p> Γ ( 1 4 ) = ( 2 π ) 3 4 ∏ k = 1 ∞ tanh ( π k 2 ) {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)=(2\pi )^{\frac {3}{4}}\prod _{k=1}^{\infty }\tanh \left({\frac {\pi k}{2}}\right)} <p>and </p> Γ ( 1 4 ) = A 3 e − G π π 2 1 6 ∏ k = 1 ∞ ( 1 − 1 2 k ) k ( − 1 ) k {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)=A^{3}e^{-{\frac {G}{\pi }}}{\sqrt {\pi }}2^{\frac {1}{6}}\prod _{k=1}^{\infty }\left(1-{\frac {1}{2k}}\right)^{k(-1)^{k}}} <p>where A is the <a href="/facts/Glaisher%25E2%2580%2593Kinkelin_constant/lK8noN4A">Glaisher–Kinkelin constant</a> and G is <a href="/facts/Catalan%2527s_constant/c9fNqG8j">Catalan's constant</a>. </p><p>The following two representations for Γ(3/4) were given by I. Mező<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> π e π 2 1 Γ ( 3 4 ) 2 = i ∑ k = − ∞ ∞ e π ( k − 2 k 2 ) θ 1 ( i π 2 ( 2 k − 1 ) , e − π ) , {\displaystyle {\sqrt {\frac {\pi {\sqrt {e^{\pi }}}}{2}}}{\frac {1}{\Gamma \left({\frac {3}{4}}\right)^{2}}}=i\sum _{k=-\infty }^{\infty }e^{\pi (k-2k^{2})}\theta _{1}\left({\frac {i\pi }{2}}(2k-1),e^{-\pi }\right),} <p>and </p> π 2 1 Γ ( 3 4 ) 2 = ∑ k = − ∞ ∞ θ 4 ( i k π , e − π ) e 2 π k 2 , {\displaystyle {\sqrt {\frac {\pi }{2}}}{\frac {1}{\Gamma \left({\frac {3}{4}}\right)^{2}}}=\sum _{k=-\infty }^{\infty }{\frac {\theta _{4}(ik\pi ,e^{-\pi })}{e^{2\pi k^{2}}}},} <p>where <i>θ</i>1 and <i>θ</i>4 are two of the <a href="/facts/Theta_function/g09xvoQQ">Jacobi theta functions</a>. </p><p>There also exist a number of <a href="/facts/Carl_Johan_Malmsten/QrPfsPCn">Malmsten integrals</a> for certain values of the gamma function:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> ∫ 1 ∞ ln ln t 1 + t 2 = π 4 ( 2 ln 2 + 3 ln π − 4 Γ ( 1 4 ) ) {\displaystyle \int _{1}^{\infty }{\frac {\ln \ln t}{1+t^{2}}}={\frac {\pi }{4}}\left(2\ln 2+3\ln \pi -4\Gamma \left({\tfrac {1}{4}}\right)\right)} ∫ 1 ∞ ln ln t 1 + t + t 2 = π 6 3 ( 8 ln 2 − 3 ln 3 + 8 ln π − 12 Γ ( 1 3 ) ) {\displaystyle \int _{1}^{\infty }{\frac {\ln \ln t}{1+t+t^{2}}}={\frac {\pi }{6{\sqrt {3}}}}\left(8\ln 2-3\ln 3+8\ln \pi -12\Gamma \left({\tfrac {1}{3}}\right)\right)} <h2 id="products">Products</h2> <p>Some product identities include: </p> ∏ r = 1 2 Γ ( r 3 ) = 2 π 3 ≈ 3.627 598 728 468 435 7012 {\displaystyle \prod _{r=1}^{2}\Gamma \left({\tfrac {r}{3}}\right)={\frac {2\pi }{\sqrt {3}}}\approx 3.627\,598\,728\,468\,435\,7012} <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A186706 ∏ r = 1 3 Γ ( r 4 ) = 2 π 3 ≈ 7.874 804 972 861 209 8721 {\displaystyle \prod _{r=1}^{3}\Gamma \left({\tfrac {r}{4}}\right)={\sqrt {2\pi ^{3}}}\approx 7.874\,804\,972\,861\,209\,8721} <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A220610 ∏ r = 1 4 Γ ( r 5 ) = 4 π 2 5 ≈ 17.655 285 081 493 524 2483 {\displaystyle \prod _{r=1}^{4}\Gamma \left({\tfrac {r}{5}}\right)={\frac {4\pi ^{2}}{\sqrt {5}}}\approx 17.655\,285\,081\,493\,524\,2483} ∏ r = 1 5 Γ ( r 6 ) = 4 π 5 3 ≈ 40.399 319 122 003 790 0785 {\displaystyle \prod _{r=1}^{5}\Gamma \left({\tfrac {r}{6}}\right)=4{\sqrt {\frac {\pi ^{5}}{3}}}\approx 40.399\,319\,122\,003\,790\,0785} ∏ r = 1 6 Γ ( r 7 ) = 8 π 3 7 ≈ 93.754 168 203 582 503 7970 {\displaystyle \prod _{r=1}^{6}\Gamma \left({\tfrac {r}{7}}\right)={\frac {8\pi ^{3}}{\sqrt {7}}}\approx 93.754\,168\,203\,582\,503\,7970} ∏ r = 1 7 Γ ( r 8 ) = 4 π 7 ≈ 219.828 778 016 957 263 6207 {\displaystyle \prod _{r=1}^{7}\Gamma \left({\tfrac {r}{8}}\right)=4{\sqrt {\pi ^{7}}}\approx 219.828\,778\,016\,957\,263\,6207} <p>In general: </p> ∏ r = 1 n Γ ( r n + 1 ) = ( 2 π ) n n + 1 {\displaystyle \prod _{r=1}^{n}\Gamma \left({\tfrac {r}{n+1}}\right)={\sqrt {\frac {(2\pi )^{n}}{n+1}}}} <p>From those products can be deduced other values, for example, from the former equations for ∏ r = 1 3 Γ ( r 4 ) {\displaystyle \prod _{r=1}^{3}\Gamma \left({\tfrac {r}{4}}\right)} , Γ ( 1 4 ) {\displaystyle \Gamma \left({\tfrac {1}{4}}\right)} and Γ ( 2 4 ) {\displaystyle \Gamma \left({\tfrac {2}{4}}\right)} , can be deduced: </p><p> Γ ( 3 4 ) = ( π 2 ) 1 4 AGM ( 2 , 1 ) 1 2 {\displaystyle \Gamma \left({\tfrac {3}{4}}\right)=\left({\tfrac {\pi }{2}}\right)^{\tfrac {1}{4}}{\operatorname {AGM} \left({\sqrt {2}},1\right)}^{\tfrac {1}{2}}} </p><p>Other rational relations include </p> Γ ( 1 5 ) Γ ( 4 15 ) Γ ( 1 3 ) Γ ( 2 15 ) = 2 3 20 5 6 5 − 7 5 + 6 − 6 5 4 {\displaystyle {\frac {\Gamma \left({\tfrac {1}{5}}\right)\Gamma \left({\tfrac {4}{15}}\right)}{\Gamma \left({\tfrac {1}{3}}\right)\Gamma \left({\tfrac {2}{15}}\right)}}={\frac {{\sqrt {2}}\,{\sqrt[{20}]{3}}}{{\sqrt[{6}]{5}}\,{\sqrt[{4}]{5-{\frac {7}{\sqrt {5}}}+{\sqrt {6-{\frac {6}{\sqrt {5}}}}}}}}}} Γ ( 1 20 ) Γ ( 9 20 ) Γ ( 3 20 ) Γ ( 7 20 ) = 5 4 ( 1 + 5 ) 2 {\displaystyle {\frac {\Gamma \left({\tfrac {1}{20}}\right)\Gamma \left({\tfrac {9}{20}}\right)}{\Gamma \left({\tfrac {3}{20}}\right)\Gamma \left({\tfrac {7}{20}}\right)}}={\frac {{\sqrt[{4}]{5}}\left(1+{\sqrt {5}}\right)}{2}}} <a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Γ ( 1 5 ) 2 Γ ( 1 10 ) Γ ( 3 10 ) = 1 + 5 2 7 10 5 4 {\displaystyle {\frac {\Gamma \left({\frac {1}{5}}\right)^{2}}{\Gamma \left({\frac {1}{10}}\right)\Gamma \left({\frac {3}{10}}\right)}}={\frac {\sqrt {1+{\sqrt {5}}}}{2^{\tfrac {7}{10}}{\sqrt[{4}]{5}}}}} <p>and many more relations for Γ(<i>n</i>/<i>d</i>) where the denominator d divides 24 or 60.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>Gamma quotients with algebraic values must be "poised" in the sense that the sum of arguments is the same (modulo 1) for the denominator and the numerator. </p><p>A more sophisticated example: </p> Γ ( 11 42 ) Γ ( 2 7 ) Γ ( 1 21 ) Γ ( 1 2 ) = 8 sin ( π 7 ) sin ( π 21 ) sin ( 4 π 21 ) sin ( 5 π 21 ) 2 1 42 3 9 28 7 1 3 {\displaystyle {\frac {\Gamma \left({\frac {11}{42}}\right)\Gamma \left({\frac {2}{7}}\right)}{\Gamma \left({\frac {1}{21}}\right)\Gamma \left({\frac {1}{2}}\right)}}={\frac {8\sin \left({\frac {\pi }{7}}\right){\sqrt {\sin \left({\frac {\pi }{21}}\right)\sin \left({\frac {4\pi }{21}}\right)\sin \left({\frac {5\pi }{21}}\right)}}}{2^{\frac {1}{42}}3^{\frac {9}{28}}7^{\frac {1}{3}}}}} <a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> <h2 id="imaginary-and-complex-arguments">Imaginary and complex arguments</h2> <p>The gamma function at the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a> <i>i</i> = √−1 gives <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A212877, <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A212878: </p> Γ ( i ) = ( − 1 + i ) ! ≈ − 0.1549 − 0.4980 i . {\displaystyle \Gamma (i)=(-1+i)!\approx -0.1549-0.4980i.} <p>It may also be given in terms of the <a href="/facts/Barnes_G-function/4bmYxJrE">Barnes G-function</a>: </p> Γ ( i ) = G ( 1 + i ) G ( i ) = e − log G ( i ) + log G ( 1 + i ) . {\displaystyle \Gamma (i)={\frac {G(1+i)}{G(i)}}=e^{-\log G(i)+\log G(1+i)}.} <p>Curiously enough, Γ ( i ) {\displaystyle \Gamma (i)} appears in the below integral evaluation:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> ∫ 0 π / 2 { cot ( x ) } d x = 1 − π 2 + i 2 log ( π sinh ( π ) Γ ( i ) 2 ) . {\displaystyle \int _{0}^{\pi /2}\{\cot(x)\}\,dx=1-{\frac {\pi }{2}}+{\frac {i}{2}}\log \left({\frac {\pi }{\sinh(\pi )\Gamma (i)^{2}}}\right).} <p>Here { ⋅ } {\displaystyle \{\cdot \}} denotes the <a href="/facts/Fractional_part/lSU6Ap1q">fractional part</a>. </p><p>Because of the <a href="/facts/Reflection_formula/aKUhTbH0">Euler Reflection Formula</a>, and the fact that Γ ( z ¯ ) = Γ ¯ ( z ) {\displaystyle \Gamma ({\bar {z}})={\bar {\Gamma }}(z)} , we have an expression for the <a href="/facts/Modulus_squared/Jzpq0R0z">modulus squared</a> of the Gamma function evaluated on the imaginary axis: </p> | Γ ( i κ ) | 2 = π κ sinh ( π κ ) {\displaystyle \left|\Gamma (i\kappa )\right|^{2}={\frac {\pi }{\kappa \sinh(\pi \kappa )}}} <p>The above integral therefore relates to the phase of Γ ( i ) {\displaystyle \Gamma (i)} . </p><p>The gamma function with other complex arguments returns </p> Γ ( 1 + i ) = i Γ ( i ) ≈ 0.498 − 0.155 i {\displaystyle \Gamma (1+i)=i\Gamma (i)\approx 0.498-0.155i} Γ ( 1 − i ) = − i Γ ( − i ) ≈ 0.498 + 0.155 i {\displaystyle \Gamma (1-i)=-i\Gamma (-i)\approx 0.498+0.155i} Γ ( 1 2 + 1 2 i ) ≈ 0.818 163 9995 − 0.763 313 8287 i {\displaystyle \Gamma ({\tfrac {1}{2}}+{\tfrac {1}{2}}i)\approx 0.818\,163\,9995-0.763\,313\,8287\,i} Γ ( 1 2 − 1 2 i ) ≈ 0.818 163 9995 + 0.763 313 8287 i {\displaystyle \Gamma ({\tfrac {1}{2}}-{\tfrac {1}{2}}i)\approx 0.818\,163\,9995+0.763\,313\,8287\,i} Γ ( 5 + 3 i ) ≈ 0.016 041 8827 − 9.433 293 2898 i {\displaystyle \Gamma (5+3i)\approx 0.016\,041\,8827-9.433\,293\,2898\,i} Γ ( 5 − 3 i ) ≈ 0.016 041 8827 + 9.433 293 2898 i . {\displaystyle \Gamma (5-3i)\approx 0.016\,041\,8827+9.433\,293\,2898\,i.} <h2 id="other-constants">Other constants</h2> <p>The gamma function has a <a href="/facts/Local_minimum/ADlgnyzV">local minimum</a> on the positive real axis </p> x min = 1.461 632 144 968 362 341 262 659 5423 … {\displaystyle x_{\min }=1.461\,632\,144\,968\,362\,341\,262\,659\,5423\ldots \,} <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A030169 <p>with the value </p> Γ ( x min ) = 0.885 603 194 410 888 700 278 815 9005 … {\displaystyle \Gamma \left(x_{\min }\right)=0.885\,603\,194\,410\,888\,700\,278\,815\,9005\ldots \,} <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A030171. <p>Integrating the <a href="/facts/Reciprocal_gamma_function/qToKveVM">reciprocal gamma function</a> along the positive real axis also gives the <a href="/facts/Frans%25C3%25A9n%25E2%2580%2593Robinson_constant/GBlrQlAq">Fransén–Robinson constant</a>. </p><p>On the negative real axis, the first local maxima and minima (zeros of the <a href="/facts/Digamma_function/nDTNu86e">digamma function</a>) are: </p> Approximate local extrema of Γ(<i>x</i>)<table><tbody><tr><th>x</th><th>Γ(<i>x</i>)</th><th><a href="/facts/OEIS/yow6cVsU">OEIS</a></th></tr><tr><td>−0.5040830082644554092582693045</td><td>−3.5446436111550050891219639933</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A175472</td></tr><tr><td>−1.5734984731623904587782860437</td><td>−2.3024072583396801358235820396</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A175473</td></tr><tr><td>−2.6107208684441446500015377157</td><td>−0.8881363584012419200955280294</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A175474</td></tr><tr><td>−3.6352933664369010978391815669</td><td>−0.2451275398343662504382300889</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A256681</td></tr><tr><td>−4.6532377617431424417145981511</td><td>−0.0527796395873194007604835708</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A256682</td></tr><tr><td>−5.6671624415568855358494741745</td><td>−0.0093245944826148505217119238</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A256683</td></tr><tr><td>−6.6784182130734267428298558886</td><td>−0.0013973966089497673013074887</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A256684</td></tr><tr><td>−7.6877883250316260374400988918</td><td>−0.0001818784449094041881014174</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A256685</td></tr><tr><td>−8.6957641638164012664887761608</td><td>−0.0000209252904465266687536973</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A256686</td></tr><tr><td>−9.7026725400018637360844267649</td><td>−0.0000021574161045228505405031</td><td><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A256687</td></tr></tbody></table> <p>The only values of <i>x</i> > 0 for which Γ(<i>x</i>) = <i>x</i> are <i>x</i> = 1 and <i>x</i> ≈ 3.5623822853908976914156443427... <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A218802. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Chowla%25E2%2580%2593Selberg_formula/R26PY2y4">Chowla–Selberg formula</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Gramain, F. (1981). "Sur le théorème de Fukagawa-Gel'fond". <i>Invent. Math</i>. 63 (3): 495–506. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1981InMat..63..495G">1981InMat..63..495G</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01389066">10.1007/BF01389066</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123079859">123079859</a>.</li> <li>Borwein, J. M.; Zucker, I. J. (1992). "Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind". <i>IMA Journal of Numerical Analysis</i>. 12 (4): 519–526. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fimanum%2F12.4.519">10.1093/imanum/12.4.519</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1186733">1186733</a>.</li> <li>X. Gourdon & P. Sebah. <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.html">Introduction to the Gamma Function</a></li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/GammaFunction.html">"Gamma Function"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li>Vidunas, Raimundas (2005). "Expressions for values of the gamma function". <i>Kyushu Journal of Mathematics</i>. 59 (2): 267–283. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.CA/0403510">math.CA/0403510</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2206%2Fkyushujm.59.267">10.2206/kyushujm.59.267</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119623635">119623635</a>.</li> <li>Vidunas, Raimundas (2005). "Expressions for values of the gamma function". <i>Kyushu J. Math</i>. 59 (2): 267–283. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0403510">math/0403510</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2206%2Fkyushujm.59.267">10.2206/kyushujm.59.267</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2188592">2188592</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119623635">119623635</a>.</li> <li>Adamchik, V. S. (2005). <a href="https://www.cs.cmu.edu/~adamchik/articles/rama.pdf">"Multiple Gamma Function and Its Application to Computation of Series"</a> (PDF). <i>The Ramanujan Journal</i>. 9 (3): 271–288. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0308074">math/0308074</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs11139-005-1868-3">10.1007/s11139-005-1868-3</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2173489">2173489</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:15670340">15670340</a>.</li> <li>Duke, W.; Imamoglu, Ö. (2006). <a href="https://www.math.ucla.edu/~wdduke/preprints/special-jntb.pdf">"Special values of multiple gamma functions"</a> (PDF). <i>Journal de Théorie des Nombres de Bordeaux</i>. 18 (1): 113–123. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.5802%2Fjtnb.536">10.5802/jtnb.536</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2245878">2245878</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Waldschmidt, Michel (2006). "Transcendence of periods: the state of the art". Pure and Applied Mathematics Quarterly. 2 (2): 435–463. doi:10.4310/PAMQ.2006.v2.n2.a3. <a href="https://hal.science/hal-00411301" target="_blank">https://hal.science/hal-00411301</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Archived copy". Retrieved 2015-03-09. <a href="https://math.stackexchange.com/q/1631760" target="_blank">https://math.stackexchange.com/q/1631760</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Mező, István (2013), "Duplication formulae involving Jacobi theta functions and Gosper's q-trigonometric functions", Proceedings of the American Mathematical Society, 141 (7): 2401–2410, doi:10.1090/s0002-9939-2013-11576-5 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Blagouchine, Iaroslav V. (2014-10-01). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. ISSN 1572-9303. <a href="https://link.springer.com/article/10.1007/s11139-013-9528-5" target="_blank">https://link.springer.com/article/10.1007/s11139-013-9528-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Weisstein, Eric W. "Gamma Function". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Raimundas Vidūnas, Expressions for Values of the Gamma Function <a href="https://arxiv.org/abs/math/0403510" target="_blank">https://arxiv.org/abs/math/0403510</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>math.stackexchange.com <a href="https://math.stackexchange.com/q/2804457" target="_blank">https://math.stackexchange.com/q/2804457</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>The webpage of István Mező <a href="https://sites.google.com/site/istvanmezo81/monthly-problems" target="_blank">https://sites.google.com/site/istvanmezo81/monthly-problems</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> </ol>