82-408: Thomas Joannes Stieltjes ( / ˈ s t iː l tʃ ə z / STEEL -chəz , Dutch: [ˈtoːmɑ ˈstiltɕəs] ; 29 December 1856 – 31 December 1894) was a Dutch mathematician . He was a pioneer in the field of moment problems and contributed to the study of continued fractions . The Thomas Stieltjes Institute for Mathematics at Leiden University , dissolved in 2011, was named after him, as
164-779: A , a ∈ ( 0 , 1 ) . {\displaystyle I=\int _{-\infty }^{\infty }{\frac {e^{ax}}{1+e^{x}}}\,dx=\int _{0}^{\infty }{\frac {v^{a-1}}{1+v}}\,dv={\frac {\pi }{\sin \pi a}},\quad a\in (0,1).} Consider the positively oriented rectangular contour C R {\displaystyle C_{R}} with vertices at R {\displaystyle R} , − R {\displaystyle -R} , R + 2 π i {\displaystyle R+2\pi i} and − R + 2 π i {\displaystyle -R+2\pi i} where R ∈ R + {\displaystyle R\in \mathbb {R} ^{+}} . Then by
246-419: A I R {\displaystyle I_{R}'=-e^{2\pi ia}I_{R}} . If A R {\displaystyle A_{R}} denotes the right vertical side of the rectangle, then | ∫ A R e a z 1 + e z d z | ≤ ∫ 0 2 π | e
328-1860: A I = − 2 π i e a π i , {\displaystyle I-e^{2\pi ia}I=-2\pi ie^{a\pi i},} from which I = π sin π a , a ∈ ( 0 , 1 ) . {\displaystyle I={\frac {\pi }{\sin \pi a}},\quad a\in (0,1).} Then Γ ( 1 − z ) = ∫ 0 ∞ e − u u − z d u = t ∫ 0 ∞ e − v t ( v t ) − z d v , t > 0 {\displaystyle \Gamma (1-z)=\int _{0}^{\infty }e^{-u}u^{-z}\,du=t\int _{0}^{\infty }e^{-vt}(vt)^{-z}\,dv,\quad t>0} and Γ ( z ) Γ ( 1 − z ) = ∫ 0 ∞ ∫ 0 ∞ e − t ( 1 + v ) v − z d v d t = ∫ 0 ∞ v − z 1 + v d v = π sin π ( 1 − z ) = π sin π z , z ∈ ( 0 , 1 ) . {\displaystyle {\begin{aligned}\Gamma (z)\Gamma (1-z)&=\int _{0}^{\infty }\int _{0}^{\infty }e^{-t(1+v)}v^{-z}\,dv\,dt\\&=\int _{0}^{\infty }{\frac {v^{-z}}{1+v}}\,dv\\&={\frac {\pi }{\sin \pi (1-z)}}\\&={\frac {\pi }{\sin \pi z}},\quad z\in (0,1).\end{aligned}}} Proving
410-488: A < 1 {\displaystyle a<1} , the integral tends to 0 {\displaystyle 0} as R → ∞ {\displaystyle R\to \infty } . Analogously, the integral over the left vertical side of the rectangle tends to 0 {\displaystyle 0} as R → ∞ {\displaystyle R\to \infty } . Therefore I − e 2 π i
492-423: A ( R + i t ) 1 + e R + i t | d t ≤ C e ( a − 1 ) R {\displaystyle \left|\int _{A_{R}}{\frac {e^{az}}{1+e^{z}}}\,dz\right|\leq \int _{0}^{2\pi }\left|{\frac {e^{a(R+it)}}{1+e^{R+it}}}\right|\,dt\leq Ce^{(a-1)R}} for some constant C {\displaystyle C} and since
574-554: A x 1 + e x d x {\displaystyle I_{R}=\int _{-R}^{R}{\frac {e^{ax}}{1+e^{x}}}\,dx} and let I R ′ {\displaystyle I_{R}'} be the analogous integral over the top side of the rectangle. Then I R → I {\displaystyle I_{R}\to I} as R → ∞ {\displaystyle R\to \infty } and I R ′ = − e 2 π i
656-456: A + bi , this product is | Γ ( a + b i ) | 2 = | Γ ( a ) | 2 ∏ k = 0 ∞ 1 1 + b 2 ( a + k ) 2 {\displaystyle |\Gamma (a+bi)|^{2}=|\Gamma (a)|^{2}\prod _{k=0}^{\infty }{\frac {1}{1+{\frac {b^{2}}{(a+k)^{2}}}}}} If
738-1388: A Gaussian integral . In general, for non-negative integer values of n {\displaystyle n} we have: Γ ( 1 2 + n ) = ( 2 n ) ! 4 n n ! π = ( 2 n − 1 ) ! ! 2 n π = ( n − 1 2 n ) n ! π Γ ( 1 2 − n ) = ( − 4 ) n n ! ( 2 n ) ! π = ( − 2 ) n ( 2 n − 1 ) ! ! π = π ( − 1 / 2 n ) n ! {\displaystyle {\begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={(2n)! \over 4^{n}n!}{\sqrt {\pi }}={\frac {(2n-1)!!}{2^{n}}}{\sqrt {\pi }}={\binom {n-{\frac {1}{2}}}{n}}n!{\sqrt {\pi }}\\[8pt]\Gamma \left({\tfrac {1}{2}}-n\right)&={(-4)^{n}n! \over (2n)!}{\sqrt {\pi }}={\frac {(-2)^{n}}{(2n-1)!!}}{\sqrt {\pi }}={\frac {\sqrt {\pi }}{{\binom {-1/2}{n}}n!}}\end{aligned}}} where
820-450: A convergent improper integral for complex numbers with positive real part: Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t , ℜ ( z ) > 0 . {\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}{\text{ d}}t,\ \qquad \Re (z)>0\,.} The gamma function then
902-477: A financial economist might study the structural reasons why a company may have a certain share price , a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock ( see: Valuation of options ; Financial modeling ). According to the Dictionary of Occupational Titles occupations in mathematics include
SECTION 10
#1732765181806984-465: A fixed integer m {\displaystyle m} , as the integer n {\displaystyle n} increases, we have that lim n → ∞ n ! ( n + 1 ) m ( n + m ) ! = 1 . {\displaystyle \lim _{n\to \infty }{\frac {n!\,\left(n+1\right)^{m}}{(n+m)!}}=1\,.} If m {\displaystyle m}
1066-618: A formula for other individual values Γ ( r ) {\displaystyle \Gamma (r)} where r {\displaystyle r} is rational, especially because according to Gauss's digamma theorem , it is possible to do so for the closely related digamma function at every rational value. However, these numbers Γ ( r ) {\displaystyle \Gamma (r)} are not known to be expressible by themselves in terms of elementary functions. It has been proved that Γ ( n + r ) {\displaystyle \Gamma (n+r)}
1148-400: A manner which will help ensure that the plans are maintained on a sound financial basis. As another example, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while
1230-464: A mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry , by deriving four corollaries to Thales's theorem . The number of known mathematicians grew when Pythagoras of Samos ( c. 582 – c. 507 BC ) established the Pythagorean school , whose doctrine it was that mathematics ruled the universe and whose motto
1312-524: A unique solution, since it allows for multiplication by any periodic function g ( x ) {\displaystyle g(x)} with g ( x ) = g ( x + 1 ) {\displaystyle g(x)=g(x+1)} and g ( 0 ) = 1 {\displaystyle g(0)=1} , such as g ( x ) = e k sin ( m π x ) {\displaystyle g(x)=e^{k\sin(m\pi x)}} . One way to resolve
1394-583: Is convex . The notation Γ ( z ) {\displaystyle \Gamma (z)} is due to Legendre . If the real part of the complex number z is strictly positive ( ℜ ( z ) > 0 {\displaystyle \Re (z)>0} ), then the integral Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t {\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt} converges absolutely , and
1476-420: Is mathematics that studies entirely abstract concepts . From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics , and at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and other applications. Another insightful view put forth is that pure mathematics
1558-451: Is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science , engineering , business , and other areas of mathematical practice. Pure mathematics
1640-518: Is a transcendental number and algebraically independent of π {\displaystyle \pi } for any integer n {\displaystyle n} and each of the fractions r = 1 6 , 1 4 , 1 3 , 2 3 , 3 4 , 5 6 {\textstyle r={\frac {1}{6}},{\frac {1}{4}},{\frac {1}{3}},{\frac {2}{3}},{\frac {3}{4}},{\frac {5}{6}}} . In general, when computing values of
1722-736: Is an entire function , converging for every complex number z . The definition for the gamma function due to Weierstrass is also valid for all complex numbers z {\displaystyle z} except non-positive integers: Γ ( z ) = e − γ z z ∏ n = 1 ∞ ( 1 + z n ) − 1 e z / n , {\displaystyle \Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}e^{z/n},} where γ ≈ 0.577216 {\displaystyle \gamma \approx 0.577216}
SECTION 20
#17327651818061804-3559: Is an entire function of order 1 {\displaystyle 1} . Since z Γ ( z ) → 1 {\displaystyle z\Gamma (z)\to 1} as z → 0 {\displaystyle z\to 0} , c 2 = 0 {\displaystyle c_{2}=0} (or an integer multiple of 2 π i {\displaystyle 2\pi i} ) and since Γ ( 1 ) = 1 {\displaystyle \Gamma (1)=1} , e − c 1 = ∏ n = 1 ∞ e − 1 n ( 1 + 1 n ) = exp ( lim N → ∞ ∑ n = 1 N ( log ( 1 + 1 n ) − 1 n ) ) = exp ( lim N → ∞ ( log ( N + 1 ) − ∑ n = 1 N 1 n ) ) = exp ( lim N → ∞ ( log N + log ( 1 + 1 N ) − ∑ n = 1 N 1 n ) ) = exp ( lim N → ∞ ( log N − ∑ n = 1 N 1 n ) ) = e − γ . {\displaystyle {\begin{aligned}e^{-c_{1}}&=\prod _{n=1}^{\infty }e^{-{\frac {1}{n}}}\left(1+{\frac {1}{n}}\right)\\&=\exp \left(\lim _{N\to \infty }\sum _{n=1}^{N}\left(\log \left(1+{\frac {1}{n}}\right)-{\frac {1}{n}}\right)\right)\\&=\exp \left(\lim _{N\to \infty }\left(\log(N+1)-\sum _{n=1}^{N}{\frac {1}{n}}\right)\right)\\&=\exp \left(\lim _{N\to \infty }\left(\log N+\log \left(1+{\frac {1}{N}}\right)-\sum _{n=1}^{N}{\frac {1}{n}}\right)\right)\\&=\exp \left(\lim _{N\to \infty }\left(\log N-\sum _{n=1}^{N}{\frac {1}{n}}\right)\right)\\&=e^{-\gamma }.\end{aligned}}} where c 1 = γ + 2 π i k {\displaystyle c_{1}=\gamma +2\pi ik} for some integer k {\displaystyle k} . Since Γ ( z ) ∈ R {\displaystyle \Gamma (z)\in \mathbb {R} } for z ∈ R ∖ Z 0 − {\displaystyle z\in \mathbb {R} \setminus \mathbb {Z} _{0}^{-}} , we have k = 0 {\displaystyle k=0} and 1 Γ ( z ) = z e γ z ∏ n = 1 ∞ e − z n ( 1 + z n ) , z ∈ C ∖ Z 0 − . {\displaystyle {\frac {1}{\Gamma (z)}}=ze^{\gamma z}\prod _{n=1}^{\infty }e^{-{\frac {z}{n}}}\left(1+{\frac {z}{n}}\right),\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}.} Equivalence of
1886-448: Is approximately the result of computing Γ ( n + 1 ) = n ! {\displaystyle \Gamma (n+1)=n!} for some large integer n {\displaystyle n} , multiplying by ( n + 1 ) z {\displaystyle (n+1)^{z}} to approximate Γ ( n + z + 1 ) {\displaystyle \Gamma (n+z+1)} , and using
1968-463: Is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles . The gamma function has no zeros, so the reciprocal gamma function 1 / Γ( z ) is an entire function . In fact, the gamma function corresponds to the Mellin transform of
2050-398: Is due to Euler, converges for all complex numbers z {\displaystyle z} except the non-positive integers, which fail because of a division by zero. Hence the above assumption produces a unique definition of z ! {\displaystyle z!} . Intuitively, this formula indicates that Γ ( z ) {\displaystyle \Gamma (z)}
2132-2107: Is even, hence 2 2 z − 1 Γ 2 ( z ) = 2 Γ ( 2 z ) ∫ 0 1 ( 1 − u 2 ) z − 1 d u . {\displaystyle 2^{2z-1}\Gamma ^{2}(z)=2\Gamma (2z)\int _{0}^{1}(1-u^{2})^{z-1}\,du.} Now assume B ( 1 2 , z ) = ∫ 0 1 t 1 2 − 1 ( 1 − t ) z − 1 d t , t = s 2 . {\displaystyle \mathrm {B} \left({\frac {1}{2}},z\right)=\int _{0}^{1}t^{{\frac {1}{2}}-1}(1-t)^{z-1}\,dt,\quad t=s^{2}.} Then B ( 1 2 , z ) = 2 ∫ 0 1 ( 1 − s 2 ) z − 1 d s = 2 ∫ 0 1 ( 1 − u 2 ) z − 1 d u . {\displaystyle \mathrm {B} \left({\frac {1}{2}},z\right)=2\int _{0}^{1}(1-s^{2})^{z-1}\,ds=2\int _{0}^{1}(1-u^{2})^{z-1}\,du.} This implies 2 2 z − 1 Γ 2 ( z ) = Γ ( 2 z ) B ( 1 2 , z ) . {\displaystyle 2^{2z-1}\Gamma ^{2}(z)=\Gamma (2z)\mathrm {B} \left({\frac {1}{2}},z\right).} Since B ( 1 2 , z ) = Γ ( 1 2 ) Γ ( z ) Γ ( z + 1 2 ) , Γ ( 1 2 ) = π , {\displaystyle \mathrm {B} \left({\frac {1}{2}},z\right)={\frac {\Gamma \left({\frac {1}{2}}\right)\Gamma (z)}{\Gamma \left(z+{\frac {1}{2}}\right)}},\quad \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }},}
2214-2797: Is known as the Euler integral of the second kind . (Euler's integral of the first kind is the beta function . ) Using integration by parts , one sees that: Γ ( z + 1 ) = ∫ 0 ∞ t z e − t d t = [ − t z e − t ] 0 ∞ + ∫ 0 ∞ z t z − 1 e − t d t = lim t → ∞ ( − t z e − t ) − ( − 0 z e − 0 ) + z ∫ 0 ∞ t z − 1 e − t d t . {\displaystyle {\begin{aligned}\Gamma (z+1)&=\int _{0}^{\infty }t^{z}e^{-t}\,dt\\&={\Bigl [}-t^{z}e^{-t}{\Bigr ]}_{0}^{\infty }+\int _{0}^{\infty }zt^{z-1}e^{-t}\,dt\\&=\lim _{t\to \infty }\left(-t^{z}e^{-t}\right)-\left(-0^{z}e^{-0}\right)+z\int _{0}^{\infty }t^{z-1}e^{-t}\,dt.\end{aligned}}} Recognizing that − t z e − t → 0 {\displaystyle -t^{z}e^{-t}\to 0} as t → ∞ , {\displaystyle t\to \infty ,} Γ ( z + 1 ) = z ∫ 0 ∞ t z − 1 e − t d t = z Γ ( z ) . {\displaystyle {\begin{aligned}\Gamma (z+1)&=z\int _{0}^{\infty }t^{z-1}e^{-t}\,dt\\&=z\Gamma (z).\end{aligned}}} Then Γ ( 1 ) {\displaystyle \Gamma (1)} can be calculated as: Γ ( 1 ) = ∫ 0 ∞ t 1 − 1 e − t d t = ∫ 0 ∞ e − t d t = 1. {\displaystyle {\begin{aligned}\Gamma (1)&=\int _{0}^{\infty }t^{1-1}e^{-t}\,dt\\&=\int _{0}^{\infty }e^{-t}\,dt\\&=1.\end{aligned}}} Thus we can show that Γ ( n ) = ( n − 1 ) ! {\displaystyle \Gamma (n)=(n-1)!} for any positive integer n by induction . Specifically,
2296-2164: Is not an integer, then this equation is meaningless, since in this section the factorial of a non-integer has not been defined yet. However, let us assume that this equation continues to hold when m {\displaystyle m} is replaced by an arbitrary complex number z {\displaystyle z} , in order to define the Gamma function for non integers: lim n → ∞ n ! ( n + 1 ) z ( n + z ) ! = 1 . {\displaystyle \lim _{n\to \infty }{\frac {n!\,\left(n+1\right)^{z}}{(n+z)!}}=1\,.} Multiplying both sides by ( z − 1 ) ! {\displaystyle (z-1)!} gives Γ ( z ) = ( z − 1 ) ! = 1 z lim n → ∞ n ! z ! ( n + z ) ! ( n + 1 ) z = 1 z lim n → ∞ ( 1 ⋅ 2 ⋯ n ) 1 ( 1 + z ) ⋯ ( n + z ) ( 2 1 ⋅ 3 2 ⋯ n + 1 n ) z = 1 z ∏ n = 1 ∞ [ 1 1 + z n ( 1 + 1 n ) z ] . {\displaystyle {\begin{aligned}\Gamma (z)&=(z-1)!\\[8pt]&={\frac {1}{z}}\lim _{n\to \infty }n!{\frac {z!}{(n+z)!}}(n+1)^{z}\\[8pt]&={\frac {1}{z}}\lim _{n\to \infty }(1\cdot 2\cdots n){\frac {1}{(1+z)\cdots (n+z)}}\left({\frac {2}{1}}\cdot {\frac {3}{2}}\cdots {\frac {n+1}{n}}\right)^{z}\\[8pt]&={\frac {1}{z}}\prod _{n=1}^{\infty }\left[{\frac {1}{1+{\frac {z}{n}}}}\left(1+{\frac {1}{n}}\right)^{z}\right].\end{aligned}}} This infinite product , which
2378-400: Is not necessarily applied mathematics : it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing
2460-509: Is not the only analytic function that extends the factorial, as one may add any analytic function that is zero on the positive integers, such as k sin ( m π x ) {\displaystyle k\sin(m\pi x)} for an integer m {\displaystyle m} . Such a function is known as a pseudogamma function , the most famous being the Hadamard function. A more restrictive requirement
2542-433: Is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems . Mathematicians are concerned with numbers , data , quantity , structure , space , models , and change . One of the earliest known mathematicians was Thales of Miletus ( c. 624 – c. 546 BC ); he has been hailed as the first true mathematician and the first known individual to whom
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2624-1330: Is the Euler–Mascheroni constant . This is the Hadamard product of 1 / Γ ( z ) {\displaystyle 1/\Gamma (z)} in a rewritten form. This definition appears in an important identity involving pi. Equivalence of the integral definition and Weierstrass definition By the integral definition, the relation Γ ( z + 1 ) = z Γ ( z ) {\displaystyle \Gamma (z+1)=z\Gamma (z)} and Hadamard factorization theorem , 1 Γ ( z ) = z e c 1 z + c 2 ∏ n = 1 ∞ e − z n ( 1 + z n ) , z ∈ C ∖ Z 0 − {\displaystyle {\frac {1}{\Gamma (z)}}=ze^{c_{1}z+c_{2}}\prod _{n=1}^{\infty }e^{-{\frac {z}{n}}}\left(1+{\frac {z}{n}}\right),\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}} for some constants c 1 , c 2 {\displaystyle c_{1},c_{2}} since 1 / Γ {\displaystyle 1/\Gamma }
2706-737: Is the Riemann–Stieltjes integral . Stieltjes was born in Zwolle on 29 December 1856. His father (who had the same first names) was a civil engineer and politician. Stieltjes Sr. was responsible for the construction of various harbours around Rotterdam , and also seated in the Dutch parliament . Stieltjes Jr. went to university at the Polytechnical School in Delft in 1873. Instead of attending lectures, he spent his student years reading
2788-474: Is the functional equation which interpolates the shifted factorial f ( n ) = ( n − 1 ) ! {\displaystyle f(n)=(n{-}1)!} : f ( x + 1 ) = x f ( x ) for any x > 0 , f ( 1 ) = 1. {\displaystyle f(x+1)=xf(x)\ {\text{ for any }}x>0,\qquad f(1)=1.} But this still does not give
2870-1774: The Legendre duplication formula Γ ( z ) Γ ( z + 1 2 ) = 2 1 − 2 z π Γ ( 2 z ) . {\displaystyle \Gamma (z)\Gamma \left(z+{\tfrac {1}{2}}\right)=2^{1-2z}\;{\sqrt {\pi }}\;\Gamma (2z).} Proof 1 With Euler's infinite product Γ ( z ) = 1 z ∏ n = 1 ∞ ( 1 + 1 / n ) z 1 + z / n {\displaystyle \Gamma (z)={\frac {1}{z}}\prod _{n=1}^{\infty }{\frac {(1+1/n)^{z}}{1+z/n}}} compute 1 Γ ( 1 − z ) Γ ( z ) = 1 ( − z ) Γ ( − z ) Γ ( z ) = z ∏ n = 1 ∞ ( 1 − z / n ) ( 1 + z / n ) ( 1 + 1 / n ) − z ( 1 + 1 / n ) z = z ∏ n = 1 ∞ ( 1 − z 2 n 2 ) = sin π z π , {\displaystyle {\frac {1}{\Gamma (1-z)\Gamma (z)}}={\frac {1}{(-z)\Gamma (-z)\Gamma (z)}}=z\prod _{n=1}^{\infty }{\frac {(1-z/n)(1+z/n)}{(1+1/n)^{-z}(1+1/n)^{z}}}=z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right)={\frac {\sin \pi z}{\pi }}\,,} where
2952-676: The Schock Prize , and the Nevanlinna Prize . The American Mathematical Society , Association for Women in Mathematics , and other mathematical societies offer several prizes aimed at increasing the representation of women and minorities in the future of mathematics. Several well known mathematicians have written autobiographies in part to explain to a general audience what it is about mathematics that has made them want to devote their lives to its study. These provide some of
3034-563: The double factorial ( 2 n − 1 ) ! ! = ( 2 n − 1 ) ( 2 n − 3 ) ⋯ ( 3 ) ( 1 ) {\displaystyle (2n-1)!!=(2n-1)(2n-3)\cdots (3)(1)} . See Particular values of the gamma function for calculated values. It might be tempting to generalize the result that Γ ( 1 2 ) = π {\textstyle \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}} by looking for
3116-675: The gamma function and elliptic functions . He became known internationally because of the Riemann–Stieltjes integral . Stieltjes' work on continued fractions earned him the Ormoy Prize (Prix Ormoy) of the Académie des Sciences in 1893. In 1884 the University of Leiden awarded him an honorary doctorate, and in 1885 he was elected to membership in the Royal Academy of Sciences of Amsterdam. In honour of Stieltjes, since 1996,
3198-478: The graduate level . In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students who pass are permitted to work on a doctoral dissertation . Mathematicians involved with solving problems with applications in real life are called applied mathematicians . Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of
3280-562: The multiplication theorem (see Eq. 5.5.6): ∏ k = 0 m − 1 Γ ( z + k m ) = ( 2 π ) m − 1 2 m 1 2 − m z Γ ( m z ) . {\displaystyle \prod _{k=0}^{m-1}\Gamma \left(z+{\frac {k}{m}}\right)=(2\pi )^{\frac {m-1}{2}}\;m^{{\frac {1}{2}}-mz}\;\Gamma (mz).} A simple but useful property, which can be seen from
3362-411: The residue theorem , ∫ C R e a z 1 + e z d z = − 2 π i e a π i . {\displaystyle \int _{C_{R}}{\frac {e^{az}}{1+e^{z}}}\,dz=-2\pi ie^{a\pi i}.} Let I R = ∫ − R R e
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3444-586: The Italian and German universities, but as they already enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment , the same influences that inspired Humboldt. The Universities of Oxford and Cambridge emphasized the importance of research , arguably more authentically implementing Humboldt's idea of a university than even German universities, which were subject to state authority. Overall, science (including mathematics) became
3526-460: The Legendre duplication formula follows: Γ ( z ) Γ ( z + 1 2 ) = 2 1 − 2 z π Γ ( 2 z ) . {\displaystyle \Gamma (z)\Gamma \left(z+{\frac {1}{2}}\right)=2^{1-2z}{\sqrt {\pi }}\;\Gamma (2z).} The duplication formula is a special case of
3608-583: The Stieltjes Prize (Stieltjesprijs) has been awarded annually for the best PhD thesis in mathematics to a student of any Dutch university. All mathematics institutes and departments of Dutch universities are asked for an overview of the PhDs that have taken place in the academic year. The list thus obtained forms the list of candidates for the prize. The award consists of a certificate and an amount of 1200 Euros . Mathematician A mathematician
3690-5069: The Weierstrass definition and Euler definition Γ ( z ) = e − γ z z ∏ n = 1 ∞ ( 1 + z n ) − 1 e z / n = 1 z lim n → ∞ e z ( log n − 1 − 1 2 − 1 3 − ⋯ − 1 n ) e z ( 1 + 1 2 + 1 3 + ⋯ + 1 n ) ( 1 + z ) ( 1 + z 2 ) ⋯ ( 1 + z n ) = 1 z lim n → ∞ 1 ( 1 + z ) ( 1 + z 2 ) ⋯ ( 1 + z n ) e z log ( n ) = lim n → ∞ n ! n z z ( z + 1 ) ⋯ ( z + n ) , z ∈ C ∖ Z 0 − {\displaystyle {\begin{aligned}\Gamma (z)&={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}e^{z/n}\\&={\frac {1}{z}}\lim _{n\to \infty }e^{z\left(\log n-1-{\frac {1}{2}}-{\frac {1}{3}}-\cdots -{\frac {1}{n}}\right)}{\frac {e^{z\left(1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}\right)}}{\left(1+z\right)\left(1+{\frac {z}{2}}\right)\cdots \left(1+{\frac {z}{n}}\right)}}\\&={\frac {1}{z}}\lim _{n\to \infty }{\frac {1}{\left(1+z\right)\left(1+{\frac {z}{2}}\right)\cdots \left(1+{\frac {z}{n}}\right)}}e^{z\log \left(n\right)}\\&=\lim _{n\to \infty }{\frac {n!n^{z}}{z(z+1)\cdots (z+n)}},\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}\end{aligned}}} Let Γ n ( z ) = n ! n z z ( z + 1 ) ⋯ ( z + n ) {\displaystyle \Gamma _{n}(z)={\frac {n!n^{z}}{z(z+1)\cdots (z+n)}}} and G n ( z ) = ( n − 1 ) ! n z z ( z + 1 ) ⋯ ( z + n − 1 ) . {\displaystyle G_{n}(z)={\frac {(n-1)!n^{z}}{z(z+1)\cdots (z+n-1)}}.} Then Γ n ( z ) = n z + n G n ( z ) {\displaystyle \Gamma _{n}(z)={\frac {n}{z+n}}G_{n}(z)} and lim n → ∞ G n + 1 ( z ) = lim n → ∞ G n ( z ) = lim n → ∞ Γ n ( z ) = Γ ( z ) , {\displaystyle \lim _{n\to \infty }G_{n+1}(z)=\lim _{n\to \infty }G_{n}(z)=\lim _{n\to \infty }\Gamma _{n}(z)=\Gamma (z),} therefore Γ ( z ) = lim n → ∞ n ! ( n + 1 ) z z ( z + 1 ) ⋯ ( z + n ) , z ∈ C ∖ Z 0 − . {\displaystyle \Gamma (z)=\lim _{n\to \infty }{\frac {n!(n+1)^{z}}{z(z+1)\cdots (z+n)}},\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}.} Then n ! ( n + 1 ) z z ( z + 1 ) ⋯ ( z + n ) = ( 2 / 1 ) z ( 3 / 2 ) z ( 4 / 3 ) z ⋯ ( ( n + 1 ) / n ) z z ( 1 + z ) ( 1 + z / 2 ) ( 1 + z / 3 ) ⋯ ( 1 + z / n ) = 1 z ∏ k = 1 n ( 1 + 1 / k ) z 1 + z / k , z ∈ C ∖ Z 0 − {\displaystyle {\frac {n!(n+1)^{z}}{z(z+1)\cdots (z+n)}}={\frac {(2/1)^{z}(3/2)^{z}(4/3)^{z}\cdots ((n+1)/n)^{z}}{z(1+z)(1+z/2)(1+z/3)\cdots (1+z/n)}}={\frac {1}{z}}\prod _{k=1}^{n}{\frac {(1+1/k)^{z}}{1+z/k}},\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}} and taking n → ∞ {\displaystyle n\to \infty } gives
3772-456: The ambiguity is the Bohr–Mollerup theorem , which shows that f ( x ) = Γ ( x ) {\displaystyle f(x)=\Gamma (x)} is the unique interpolating function for the factorial, defined over the positive reals, which is logarithmically convex , meaning that y = log f ( x ) {\displaystyle y=\log f(x)}
3854-583: The base case is that Γ ( 1 ) = 1 = 0 ! {\displaystyle \Gamma (1)=1=0!} , and the induction step is that Γ ( n + 1 ) = n Γ ( n ) = n ( n − 1 ) ! = n ! . {\displaystyle \Gamma (n+1)=n\Gamma (n)=n(n-1)!=n!.} The identity Γ ( z ) = Γ ( z + 1 ) z {\textstyle \Gamma (z)={\frac {\Gamma (z+1)}{z}}} can be used (or, yielding
3936-445: The best glimpses into what it means to be a mathematician. The following list contains some works that are not autobiographies, but rather essays on mathematics and mathematicians with strong autobiographical elements. Gamma function In mathematics , the gamma function (represented by Γ, capital Greek letter gamma ) is the most common extension of the factorial function to complex numbers . Derived by Daniel Bernoulli ,
4018-1061: The desired result. Besides the fundamental property discussed above: Γ ( z + 1 ) = z Γ ( z ) {\displaystyle \Gamma (z+1)=z\ \Gamma (z)} other important functional equations for the gamma function are Euler's reflection formula Γ ( 1 − z ) Γ ( z ) = π sin π z , z ∉ Z {\displaystyle \Gamma (1-z)\Gamma (z)={\frac {\pi }{\sin \pi z}},\qquad z\not \in \mathbb {Z} } which implies Γ ( z − n ) = ( − 1 ) n − 1 Γ ( − z ) Γ ( 1 + z ) Γ ( n + 1 − z ) , n ∈ Z {\displaystyle \Gamma (z-n)=(-1)^{n-1}\;{\frac {\Gamma (-z)\Gamma (1+z)}{\Gamma (n+1-z)}},\qquad n\in \mathbb {Z} } and
4100-451: The factorial, x ! = 1 × 2 × ⋯ × x is only valid when x is a positive integer, and no elementary function has this property, but a good solution is the gamma function f ( x ) = Γ ( x + 1 ) {\displaystyle f(x)=\Gamma (x+1)} . The gamma function is not only smooth but analytic (except at the non-positive integers), and it can be defined in several explicit ways. However, it
4182-540: The fields of probability , statistics , analytic number theory , and combinatorics . The gamma function can be seen as a solution to the interpolation problem of finding a smooth curve y = f ( x ) {\displaystyle y=f(x)} that connects the points of the factorial sequence: ( x , y ) = ( n , n ! ) {\displaystyle (x,y)=(n,n!)} for all positive integer values of n {\displaystyle n} . The simple formula for
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#17327651818064264-500: The focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. The German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of
4346-1060: The following. There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics or physics. Prominent prizes in mathematics include the Abel Prize , the Chern Medal , the Fields Medal , the Gauss Prize , the Nemmers Prize , the Balzan Prize , the Crafoord Prize , the Shaw Prize , the Steele Prize , the Wolf Prize ,
4428-474: The gamma function Γ ( z ) {\displaystyle \Gamma (z)} is defined for all complex numbers z {\displaystyle z} except non-positive integers, and for every positive integer z = n {\displaystyle z=n} , Γ ( n ) = ( n − 1 ) ! . {\displaystyle \Gamma (n)=(n-1)!\,.} The gamma function can be defined via
4510-633: The imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models . Mathematicians and applied mathematicians are considered to be two of the STEM (science, technology, engineering, and mathematics) careers. The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics"
4592-580: The kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of "freedom of scientific research, teaching and study." Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education , and then proceed to specialize in topics of their own choice at
4674-474: The king of Prussia , Fredrick William III , to build a university in Berlin based on Friedrich Schleiermacher 's liberal ideas; the goal was to demonstrate the process of the discovery of knowledge and to teach students to "take account of fundamental laws of science in all their thinking." Thus, seminars and laboratories started to evolve. British universities of this period adopted some approaches familiar to
4756-464: The last equality is a known result . A similar derivation begins with Weierstrass's definition. Proof 2 First prove that I = ∫ − ∞ ∞ e a x 1 + e x d x = ∫ 0 ∞ v a − 1 1 + v d v = π sin π
4838-437: The limit definition, is: Γ ( z ) ¯ = Γ ( z ¯ ) ⇒ Γ ( z ) Γ ( z ¯ ) ∈ R . {\displaystyle {\overline {\Gamma (z)}}=\Gamma ({\overline {z}})\;\Rightarrow \;\Gamma (z)\Gamma ({\overline {z}})\in \mathbb {R} .} In particular, with z =
4920-429: The negative exponential function : Γ ( z ) = M { e − x } ( z ) . {\displaystyle \Gamma (z)={\mathcal {M}}\{e^{-x}\}(z)\,.} Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It appears as a factor in various probability-distribution functions and other formulas in
5002-531: The probability and likely cost of the occurrence of an event such as death, sickness, injury, disability, or loss of property. Actuaries also address financial questions, including those involving the level of pension contributions required to produce a certain retirement income and the way in which a company should invest resources to maximize its return on investments in light of potential risk. Using their broad knowledge, actuaries help design and price insurance policies, pension plans, and other financial strategies in
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#17327651818065084-2546: The real part is an integer or a half-integer, this can be finitely expressed in closed form : | Γ ( b i ) | 2 = π b sinh π b | Γ ( 1 2 + b i ) | 2 = π cosh π b | Γ ( 1 + b i ) | 2 = π b sinh π b | Γ ( 1 + n + b i ) | 2 = π b sinh π b ∏ k = 1 n ( k 2 + b 2 ) , n ∈ N | Γ ( − n + b i ) | 2 = π b sinh π b ∏ k = 1 n ( k 2 + b 2 ) − 1 , n ∈ N | Γ ( 1 2 ± n + b i ) | 2 = π cosh π b ∏ k = 1 n ( ( k − 1 2 ) 2 + b 2 ) ± 1 , n ∈ N {\displaystyle {\begin{aligned}|\Gamma (bi)|^{2}&={\frac {\pi }{b\sinh \pi b}}\\[1ex]\left|\Gamma \left({\tfrac {1}{2}}+bi\right)\right|^{2}&={\frac {\pi }{\cosh \pi b}}\\[1ex]\left|\Gamma \left(1+bi\right)\right|^{2}&={\frac {\pi b}{\sinh \pi b}}\\[1ex]\left|\Gamma \left(1+n+bi\right)\right|^{2}&={\frac {\pi b}{\sinh \pi b}}\prod _{k=1}^{n}\left(k^{2}+b^{2}\right),\quad n\in \mathbb {N} \\[1ex]\left|\Gamma \left(-n+bi\right)\right|^{2}&={\frac {\pi }{b\sinh \pi b}}\prod _{k=1}^{n}\left(k^{2}+b^{2}\right)^{-1},\quad n\in \mathbb {N} \\[1ex]\left|\Gamma \left({\tfrac {1}{2}}\pm n+bi\right)\right|^{2}&={\frac {\pi }{\cosh \pi b}}\prod _{k=1}^{n}\left(\left(k-{\tfrac {1}{2}}\right)^{2}+b^{2}\right)^{\pm 1},\quad n\in \mathbb {N} \\[-1ex]&\end{aligned}}} First, consider
5166-507: The real part is integer or half-integer quickly follow by induction using the recurrence relation in the positive and negative directions. Perhaps the best-known value of the gamma function at a non-integer argument is Γ ( 1 2 ) = π , {\displaystyle \Gamma \left({\tfrac {1}{2}}\right)={\sqrt {\pi }},} which can be found by setting z = 1 2 {\textstyle z={\frac {1}{2}}} in
5248-484: The real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Many professional mathematicians also engage in the teaching of mathematics. Duties may include: Many careers in mathematics outside of universities involve consulting. For instance, actuaries assemble and analyze data to estimate
5330-414: The reciprocal 1 Γ ( z ) = z ∏ n = 1 ∞ [ ( 1 + z n ) / ( 1 + 1 n ) z ] {\displaystyle {\frac {1}{\Gamma (z)}}=z\prod _{n=1}^{\infty }\left[\left(1+{\frac {z}{n}}\right)/{\left(1+{\frac {1}{n}}\right)^{z}}\right]}
5412-1034: The reflection formula applied to z = 1 2 + b i {\displaystyle z={\tfrac {1}{2}}+bi} . Γ ( 1 2 + b i ) Γ ( 1 − ( 1 2 + b i ) ) = Γ ( 1 2 + b i ) Γ ( 1 2 − b i ) = π sin π ( 1 2 + b i ) = π cos π b i = π cosh π b {\displaystyle \Gamma ({\tfrac {1}{2}}+bi)\Gamma \left(1-({\tfrac {1}{2}}+bi)\right)=\Gamma ({\tfrac {1}{2}}+bi)\Gamma ({\tfrac {1}{2}}-bi)={\frac {\pi }{\sin \pi ({\tfrac {1}{2}}+bi)}}={\frac {\pi }{\cos \pi bi}}={\frac {\pi }{\cosh \pi b}}} Formulas for other values of z {\displaystyle z} for which
5494-1114: The reflection formula applied to z = b i {\displaystyle z=bi} . Γ ( b i ) Γ ( 1 − b i ) = π sin π b i {\displaystyle \Gamma (bi)\Gamma (1-bi)={\frac {\pi }{\sin \pi bi}}} Applying the recurrence relation to the second term: − b i ⋅ Γ ( b i ) Γ ( − b i ) = π sin π b i {\displaystyle -bi\cdot \Gamma (bi)\Gamma (-bi)={\frac {\pi }{\sin \pi bi}}} which with simple rearrangement gives Γ ( b i ) Γ ( − b i ) = π − b i sin π b i = π b sinh π b {\displaystyle \Gamma (bi)\Gamma (-bi)={\frac {\pi }{-bi\sin \pi bi}}={\frac {\pi }{b\sinh \pi b}}} Second, consider
5576-1408: The reflection formula for all z ∈ ( 0 , 1 ) {\displaystyle z\in (0,1)} proves it for all z ∈ C ∖ Z {\displaystyle z\in \mathbb {C} \setminus \mathbb {Z} } by analytic continuation. The beta function can be represented as B ( z 1 , z 2 ) = Γ ( z 1 ) Γ ( z 2 ) Γ ( z 1 + z 2 ) = ∫ 0 1 t z 1 − 1 ( 1 − t ) z 2 − 1 d t . {\displaystyle \mathrm {B} (z_{1},z_{2})={\frac {\Gamma (z_{1})\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt.} Setting z 1 = z 2 = z {\displaystyle z_{1}=z_{2}=z} yields Γ 2 ( z ) Γ ( 2 z ) = ∫ 0 1 t z − 1 ( 1 − t ) z − 1 d t . {\displaystyle {\frac {\Gamma ^{2}(z)}{\Gamma (2z)}}=\int _{0}^{1}t^{z-1}(1-t)^{z-1}\,dt.} After
5658-405: The reflection or duplication formulas, by using the relation to the beta function given below with z 1 = z 2 = 1 2 {\textstyle z_{1}=z_{2}={\frac {1}{2}}} , or simply by making the substitution u = z {\displaystyle u={\sqrt {z}}} in the integral definition of the gamma function, resulting in
5740-490: The relationship Γ ( x + 1 ) = x Γ ( x ) {\displaystyle \Gamma (x+1)=x\Gamma (x)} backwards n + 1 {\displaystyle n+1} times to get an approximation for Γ ( z ) {\displaystyle \Gamma (z)} ; and furthermore that this approximation becomes exact as n {\displaystyle n} increases to infinity. The infinite product for
5822-670: The required diplomas. In 1884, Hermite and professor David Bierens de Haan arranged for an honorary doctorate to be granted to Stieltjes by Leiden University , enabling him to become a professor. In 1885, he was appointed as member of the Royal Dutch Academy of Sciences (Koninklijke Nederlandse Akademie van Wetenschappen, KNAW), and the next year he became a foreign member. In 1889, he was appointed professor of differential and integral calculus at Toulouse University . Stieltjes died on 31 December 1894 in Toulouse , France. He
5904-647: The rest of his life. He originally wrote to Hermite concerning celestial mechanics , but the subject quickly turned to mathematics and he began to devote his spare time to mathematical research. The director of Leiden Observatory, van de Sande-Bakhuyzen, responded quickly to Stieltjes' request on 1 January 1883 to stop his observational work to allow him to work more on mathematical topics. In 1883, he also married Elizabeth Intveld in May. She also encouraged him to move from astronomy to mathematics. And in September, Stieltjes
5986-418: The same result, analytic continuation can be used) to uniquely extend the integral formulation for Γ ( z ) {\displaystyle \Gamma (z)} to a meromorphic function defined for all complex numbers z , except integers less than or equal to zero. It is this extended version that is commonly referred to as the gamma function. There are many equivalent definitions. For
6068-403: The seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle , and at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics . Moving into the 19th century, the objective of universities all across Europe evolved from teaching the "regurgitation of knowledge" to "encourag[ing] productive thinking." In 1810, Alexander von Humboldt convinced
6150-698: The substitution t = 1 + u 2 {\displaystyle t={\frac {1+u}{2}}} : Γ 2 ( z ) Γ ( 2 z ) = 1 2 2 z − 1 ∫ − 1 1 ( 1 − u 2 ) z − 1 d u . {\displaystyle {\frac {\Gamma ^{2}(z)}{\Gamma (2z)}}={\frac {1}{2^{2z-1}}}\int _{-1}^{1}\left(1-u^{2}\right)^{z-1}\,du.} The function ( 1 − u 2 ) z − 1 {\displaystyle (1-u^{2})^{z-1}}
6232-450: The works of Gauss and Jacobi — the consequence of this being he failed his examinations. There were two further failures (in 1875 and 1876), and his father despaired. His father was friends with H. G. van de Sande Bakhuyzen (who was the director of Leiden University ), and Stieltjes Jr. was able to get a job as an assistant at Leiden Observatory . Soon afterwards, Stieltjes began a correspondence with Charles Hermite which lasted for
6314-943: Was Al-Khawarizmi . A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics , maths and astronomy of Ibn al-Haytham . The Renaissance brought an increased emphasis on mathematics and science to Europe. During this period of transition from a mainly feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli (founder of accounting ); Niccolò Fontana Tartaglia (notable engineer and bookkeeper); Gerolamo Cardano (earliest founder of probability and binomial expansion); Robert Recorde (physician) and François Viète (lawyer). As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in
6396-697: Was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria ( c. AD 350 – 415). She succeeded her father as librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria punished her, presuming she
6478-517: Was asked to substitute at University of Delft for F.J. van den Berg. From then until December of that year, he lectured on analytical geometry and on descriptive geometry . He resigned his post at the observatory at the end of that year. In 1884, Stieltjes applied for a chair in Groningen . He was initially accepted, but in the end turned down by the Department of Education, since he lacked
6560-598: Was buried in Terre-Cabade cemetery [ fr ] on 2 January 1895. Stieltjes worked on almost all branches of analysis , continued fractions and number theory . For his work, he is sometimes referred to as " the father of the analytic theory of continued fractions ". His work is also seen as important as a first step towards the theory of Hilbert spaces . Other important contributions to mathematics that he made involved discontinuous functions and divergent series , differential equations , interpolation ,
6642-585: Was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles). Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages
6724-431: Was ongoing throughout the reign of certain caliphs, and it turned out that certain scholars became experts in the works they translated, and in turn received further support for continuing to develop certain sciences. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support
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