Remark 1. &=& \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. The partial derivative of a characteristic function (exercise). &=& -\frac{\pi}{2}\log(4)=-\pi\log(2). $$ Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Only a tiny insight in the Gamma function. -\log(4)\Gamma(2z+1)4^{-z}\Gamma^{-2}(z+1)\}. How is this done? 2\int_0^{\pi/2}\log(\sin(x))\,dx&=&\frac{\pi}{2}(-2\gamma+2\gamma-\log(4))\\ You look at some specific $x$. $$ Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. \int^{\infty}_{0} e^{-t} ln(t) t^{z-1} dt$$ Connect and share knowledge within a single location that is structured and easy to search. Use MathJax to format equations. (When z is a natural number, (z) =(z-1)! rev2022.12.9.43105. (Abramowitz and Stegun (1965, p. \end{eqnarray} I dont know exactly what Eulers thought process was, but he is the one who discovered the natural number e, so he must have experimented a lot with multiplying e with other functions to find the current form. as the dominating function. [6], [7] used the neutrix Once you have sufficient, provide answers that don't require clarification from the asker, Help us identify new roles for community members, Prove $(n-1)! the codes of Gamma function (mostly Lanczos approximation) in 60+ different language - C, C++, C#, python, java, etc. }{2 \Gamma(n+3/2)} This is one of the many definitions of the Euler-Mascheroni constant. trigamma uses an asymptotic expansion where Re(x) > 5 and a recurrence formula to such a case where Re(x) <= 5. Consider just two of the provably equivalent definitions of the Beta function: B(x, y) = 2 / 2 0 sin(t)2x 1cos(t)2y 1dt = (x)(y) (x + y). What's the next step? Of course, series for higher derivatives are given by repeated dierentiation. Try it and let me know if you find an interesting way to do so! You pick x 0, x 1 so that 0 < x 0 < x < x 1 < + . Read free for 30 days with the inequality $0\leq \log(t)\leq\sqrt{t}$ for $t\geq 1$ to prove that the hypothesis of the dominated convergence theorem are fulfilled, hence we may differentiate under the integral sign. From Reciprocal times Derivative of Gamma Function: Electromagnetic radiation and black body radiation, What does a light wave look like? Use MathJax to format equations. Would it be possible, given current technology, ten years, and an infinite amount of money, to construct a 7,000 foot (2200 meter) aircraft carrier? The digamma function is often denoted as or [3] (the uppercase form of the archaic Greek consonant digamma meaning double-gamma ). $$\Gamma^{\prime}(z) = \frac{d}{dz} \int^{\infty}_{0} e^{-t}t^{z-1}dt = \int^{\infty}_{0} \frac{d}{dz} e^{-t}t^{z-1}dt = First math video on this channel! It only takes a minute to sign up. $$ Answer (1 of 3): The antiderivative cannot be expressed in elementary functions, as others have shown, but that won't stop us from finding it nonetheless. So we have that Here is a quick look at the graph of the Gamma function in real numbers. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, The Digamma function is in relation to the gamma function. So To subscribe to this RSS feed, copy and paste this URL into your RSS reader. \log(\sin(x)) \ \mathrm{d}x = \frac{\pi}{2}\left(-2\gamma+2\gamma-\log(4)\right) = -\frac{\pi}{2}\log(4) = -\pi\log(2) Counterexamples to differentiation under integral sign, revisited, i2c_arm bus initialization and device-tree overlay. Connecting three parallel LED strips to the same power supply. Consider just two of the provably equivalent definitions of the Beta function: What properties should my fictional HEAT rounds have to punch through heavy armor and ERA? B(n+1,\frac{1}{2}): \int_0^{\pi/2}\sin^{2n+1}(x)\,dx=\frac{\sqrt{\pi} \cdot n! How is the merkle root verified if the mempools may be different? 38,938 Solution 1. +2\Gamma(2z+1)4^{-z}\Gamma^{-3}(z+1)\Gamma'(z+1)\\ $$ Thanks for contributing an answer to Mathematics Stack Exchange! Also, it has automatically delivered the fact that (z) 6= 0 . Hence, $$ digamma Function is basically, digamma (x) = d (ln (factorial (n-1)))/dx Syntax: digamma (x) Parameters: x: Numeric vector Example 1: # R program to find logarithmic derivative # of the gamma value Is there any reason on passenger airliners not to have a physical lock between throttles? The gamma function has no zeroes, so the reciprocal gamma function1/(z)is an entire function. Thanks for contributing an answer to Mathematics Stack Exchange! Because the Gamma function extends the factorial function, it satisfies a recursion relation. Central limit theorem replacing radical n with n, MOSFET is getting very hot at high frequency PWM. Can you calculate (4.8) by hand? Correctly formulate Figure caption: refer the reader to the web version of the paper? If you take one thing away from this post, it should be this section. EMILY GULLERUD, michael vaughan The Gamma Function May 10, 2017 9 / 15 \end{eqnarray} for real numbers until. $$ \int^{\pi/2}_0 \! Derivative of factorial when we have summation in the factorial? \frac{\int_0^{\pi/2}\sin^{2\cdot 0}(x)\,dx}{\int_0^{\pi/2}\sin^{2\cdot 0+1}x\,dx}=\frac{\pi/2}{1}=\frac{\pi}{2}. \sin^{2z}(x) \log(\sin(x)) \ \mathrm{d}x = I had actually got $\displaystyle\int_0^{\pi/2}\sin^{2z}(x)dx = \frac{\Gamma(z+\frac{1}{2})}{\Gamma(z+1)}\frac{\sqrt\pi}{2}$ instead. \begin{align} Disconnect vertical tab connector from PCB. Gamma function also appears in the general formula for the volume of an n-sphere. Do bracers of armor stack with magic armor enhancements and special abilities? At what point in the prequels is it revealed that Palpatine is Darth Sidious? Was the ZX Spectrum used for number crunching? Irreducible representations of a product of two groups, PSE Advent Calendar 2022 (Day 11): The other side of Christmas. Where does the idea of selling dragon parts come from? $$ Sorry but I don't see it we have $0 dt \quad \quad x>0$$, I.e. $$ Plot it yourself and see how z changes the shape of the Gamma function! What does my answer mean? \int^{\infty}_{0} e^{-t} \frac{d}{dz} t^{z-1} dt = $$, $$ where $\psi$ is the digamma function. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. If you have Im an Engineering Manager at Scale AI and this is my notepad for Applied Math / CS / Deep Learning topics. The factorial function is defined only for discrete points (for positive integers black dots in the graph above), but we wanted to connect the black dots. For the proof addicts: Lets prove the red arrow above. In order to start this off, we apply the definition of the digamma function: \displaystyle \frac{\Gamma'(z)}{\Gamma(z)} = \psi(z). \int^{\infty}_{0} e^{-t} \frac{d}{dz} t^{z-1} dt = Hence, Now differentiate both sides with respect to $z$ which yields, $$ Then: $\map {\Gamma'} 1 = -\gamma$ where: $\map {\Gamma'} 1$ denotes the derivative of the Gamma function evaluated at $1$ $\gamma$ denotes the Euler-Mascheroni constant. Connect and share knowledge within a single location that is structured and easy to search. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. \begin{eqnarray} Was the ZX Spectrum used for number crunching? \Gamma'(1) = - \gamma = \int_{0}^{\infty} e^{-t} \, \ln(t) \, dt. $$ $$ $$, Finally set $z=0$ and note that $\Gamma'(1)=-\gamma$ to complete the integration: Show that $\Gamma^{(n)}(z) = \int_0^\infty t^{z-1}(\log(t))^ne^{-t}dt$, Prove $\int_{-\infty}^{\infty} e^{2x}x^2 e^{-e^{x}}dx=\gamma^2 -2\gamma+\zeta(2)$. Help us identify new roles for community members, The right way to find $\frac{d}{ds}\Gamma (s)$. $$ $$ Can you implement this integral from 0 to infinity adding the term infinite times programmatically? Is there something special in the visible part of electromagnetic spectrum? }\\ Asking for help, clarification, or responding to other answers. 2\int_0^{\pi/2}\sin^{2z}(x)\log(\sin(x))\,dx =\frac{\pi}{2} \{2\Gamma'(2z+1)4^{-z}\Gamma^{-2}(z+1)\\ (Abramowitz and Stegun (1965, p. $$ Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Can anybody tell me if I'm on the right track? Many probability distributions are defined by using the gamma function such as Gamma distribution, Beta distribution, Dirichlet distribution, Chi-squared distribution, and Student's t-distribution, etc. Use MathJax to format equations. $\psi(x)=\frac{d}{dx}\log(\Gamma(x))$, http://www.wolframalpha.com/input/?i=integrate+log%28sin%28x%29%29+from+x%3D0+to+x%3Dpi%2F2, Help us identify new roles for community members, Solve the integral $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$, Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$, Closed form of $\int_0^{\pi/2} \frac{\arctan^2 (\sin^2 \theta)}{\sin^2 \theta}\,d\theta$, Proving a generalisation of the integral $\int_0^\infty\frac{\sin(x)}{x}dx$, Relation between integral, gamma function, elliptic integral, and AGM, Is there another way of evaluating $\lim_{x \to 0} \Gamma(x)(\gamma+\psi(1+x))=\frac{\pi^2}{6}$, Integral $\int_0^1 \sqrt{\frac{x^2-2}{x^2-1}}\, dx=\frac{\pi\sqrt{2\pi}}{\Gamma^2(1/4)}+\frac{\Gamma^2(1/4)}{4\sqrt{2\pi}}$. We are going to prove this shortly.). (= (5) = 24) as we expected. \end{align} \begin{align} To learn more, see our tips on writing great answers. An excellent discussion of this topic can be found in the book The Gamma Function by James Bonnar. (If you are interested in solving it by hand, here is a good starting point.). Use logo of university in a presentation of work done elsewhere. Asking for help, clarification, or responding to other answers. (Are you working on something today that will be used 300 years later?;). 258.) Could an oscillator at a high enough frequency produce light instead of radio waves? Does balls to the wall mean full speed ahead or full speed ahead and nosedive? Consider just two of the provably equivalent definitions of the Beta function: Many probability distributions are defined by using the gamma function such as Gamma distribution, Beta distribution, Dirichlet distribution, Chi-squared distribution, and Students t-distribution, etc.For data scientists, machine learning engineers, researchers, the Gamma function is probably one of the most widely used functions because it is employed in many distributions. Is it possible to exchange the derivative sign with the integral sign in $\;\frac{d}{dy}(\int_0^\infty F(x)\frac{e^{-x/y}}{y}\,dx)\;$? The rubber protection cover does not pass through the hole in the rim. Hence the quotient of these two integrals is In the United States, must state courts follow rulings by federal courts of appeals? Effect of coal and natural gas burning on particulate matter pollution. 258.) Two of the most often used implementations are Stirlings approximation and Lanczos approximation. Accuracy is good. $$ \int_{0}^{1}t^{x-1}\log(t)\,dx = -\frac{1}{x^2}\qquad (x>0) $$ \int_0^{\pi/2}\log(\sin(x))\,dx=-\frac{\pi}{2}\log(2). However, there are some mistakes expressed in Theorem 4, 5 in [2] and the corresponding corrections will be shown in Remark 2.4 and 2.5 in this paper. rev2022.12.9.43105. \sin^{2z} (x) \ \mathrm{d}x = \frac{\pi}{2}\Gamma(2z+1)4^{-z}\Gamma^{-2}(z+1) It was introduced by the famous mathematician L. Euler (1729) as a natural extension of the factorial operation from positive integers to real and even complex values of this argument. \frac{\pi}{2}&\left\{ 2\psi(2z+1)4^{-z}\Gamma^{-2}(z+1) \right. (3D model). \begin{eqnarray} Is energy "equal" to the curvature of spacetime? -\log(n))=0$, Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n! Effect of coal and natural gas burning on particulate matter pollution. How would you solve the integrationabove? 2\int_0^{\pi/2}\sin^{2z}(x)\log(\sin(x))\,dx =\frac{\pi}{2} \{2\Gamma'(2z+1)4^{-z}\Gamma^{-2}(z+1)\\ What is this fallacy: Perfection is impossible, therefore imperfection should be overlooked. $$, Finally set $z=0$ and note that $\Gamma'(1)=-\gamma$ to complete the integration: Later, because of its great importance, it was studied by other eminent . You pick $x_0,x_1$ so that $0 < x_0 < x < x_1 < +\infty$. $$ After my calculations I ended up with: Consider the integral form of the Gamma function, Therefore, if you understand the Gamma function well, you will have a better understanding of a lot of applications in which it appears! Hence the quotient of these two integrals is The gamma function is defined as an integral from zero to infinity. $$ We want to extend the factorial function to all complex numbers. The best answers are voted up and rise to the top, Not the answer you're looking for? $$ It only takes a minute to sign up. What definition the the gamma function are you using? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Do non-Segwit nodes reject Segwit transactions with invalid signature? https://github.com/aerinkim/TowardsDataScience/blob/master/Gamma%20Function.ipynb. Regarding the two expressions and your doubt about their equality: The equality of $\displaystyle \frac{\Gamma(2z+1)}{4^z\Gamma^2(z+1)}\frac{\pi}{2}$ and $\displaystyle \frac{\Gamma(z+\frac{1}{2})}{\Gamma(z+1)}\frac{\sqrt\pi}{2}$ can be shown by using the fact that $\Gamma(z)\Gamma(z+\frac12)=2^{1-2z}\sqrt{\pi}\Gamma(2z)$ (see wiki): $$\Gamma(2z+1)=\Gamma\left(2\left(z+\frac12\right)\right) = \frac{\Gamma\left(z+\frac12\right)\Gamma\left(z+1\right)}{2^{-2z}\sqrt{\pi}}$$, Thus, Viewed 3k times. Gamma Distribution Intuition and Derivation. - Mariana Mar 10, 2021 at 18:56 Add a comment 1 Answer Sorted by: 4 First note that by definition of the polygamma function: ( ) ( ) = 2 log ( ) = ( 1) ( ). Why is the overall charge of an ionic compound zero? $$ Making statements based on opinion; back them up with references or personal experience. The Gamma function connects the black dots and draws the curve nicely. \end{eqnarray} 17.837 falls between 3! + 1 = n^2$ has only one integer solution, How to find the formula for $\Gamma^{\prime}(m) \textrm{ and }\Gamma^{\prime \prime}(m)?$, $\lim_{(x\pi/6)}\frac{2\log((\sin x))-\log}{(\sec 2x)-1}$. An interesting side note: Euler became blind at age 64 however he produced almost half of his total works after losing his sight. Central limit theorem replacing radical n with n. Tabularray table when is wraped by a tcolorbox spreads inside right margin overrides page borders. trigamma (x) calculates the second derivatives of the logarithm of the gamma function. $$ \int_0^{\pi/2}\sin^{2z}(x)\,dx=\frac{\pi}{2}\frac{\Gamma(2z+1)}{4^z \Gamma^2(z+1)}=\frac{\pi}{2}\Gamma(2z+1)4^{-z}\Gamma^{-2}(z+1). In general it holds that: d d x ( s, x) = x s 1 e x. $$ $$, $$ $$ We also have the formulas. Is it cheating if the proctor gives a student the answer key by mistake and the student doesn't report it? Euler's limit denes the gamma function for all zexcept negative integers, whereas the integral denition only applies for Re z>0. How did the Gamma function end up with current terms x^z and e^-x? So @Jonathen Look up "differentiation under the integral sign". \sin^{2z} (x) \ \mathrm{d}x = \frac{\pi}{2}\Gamma(2z+1)4^{-z}\Gamma^{-2}(z+1) $$ then differentiating both sides with respect to $z$ gives \begin{align} (Notice the intersection at positive integers because sin(z) is zero!) \frac{\int_0^{\pi/2}\sin^{2\cdot 0}(x)\,dx}{\int_0^{\pi/2}\sin^{2\cdot 0+1}x\,dx}=\frac{\pi/2}{1}=\frac{\pi}{2}. Yes, I can find the derivative of digamma (a.k.a trigamma function) is Var (logW), where W ~ Gamma ( ,1). \int_0^{\pi/2}\sin^{2z}(x)\,dx=\frac{\pi}{2}\frac{\Gamma(2z+1)}{4^z \Gamma^2(z+1)}=\frac{\pi}{2}\Gamma(2z+1)4^{-z}\Gamma^{-2}(z+1). But I am guessing they are equivalent and differentiating them would use the same technique. \sin^{2z}(x) \log(\sin(x)) \ \mathrm{d}x = Did the apostolic or early church fathers acknowledge Papal infallibility? Then the above dominates for all $y \in (x_0,x_1)$. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. Please, This does not provide an answer to the question. $$ For me (and many others so far), there is no quick and easy way to evaluate the Gamma function of fractions manually. For x 0 < x < x 1, take. So we have that You do it locally. We conclude that 2\int^{\pi/2}_0 \! The Gamma function, (z) in blue, plotted along with (z) + sin(z) in green. The log-gamma function The Gamma function grows rapidly, so taking the natural logarithm yields a function which grows much more slowly: ln( z) = ln( z + 1) lnz This function is used in many computing environments and in the context of wave propogation. Are the S&P 500 and Dow Jones Industrial Average securities? Did neanderthals need vitamin C from the diet? as confirmed by wolfram, http://www.wolframalpha.com/input/?i=integrate+log%28sin%28x%29%29+from+x%3D0+to+x%3Dpi%2F2. It only takes a minute to sign up. Both are valid analytic continuations of the factorials to the non-integers. $$ How do you prove that Since differentiability is a local property, for the derivative at $x$ it is irrelevant what happens outside $(x_0,x_1)$. Is it cheating if the proctor gives a student the answer key by mistake and the student doesn't report it? MathJax reference. &=& \frac{2n+1}{2n}\frac{2n-1}{2n}\frac{2n-1}{2n-2}\cdots\frac{3}{4}\frac{3}{2}\frac{1}{2}\frac{\pi}{2} digamma (x) calculates the digamma function which is the logarithmic derivative of the gamma function, (x) = d (ln ( (x)))/dx = ' (x)/ (x). $$ \begin{eqnarray} Hebrews 1:3 What is the Relationship Between Jesus and The Word of His Power? The digamma function is the derivative of the log gamma function. \Gamma'(x) = \int_{0}^{\infty} e^{-t} \, t^{x-1} \, \ln(t) \, dt. Trying to prove that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} How is the merkle root verified if the mempools may be different? \int^{\pi/2}_0 \! Why does my stock Samsung Galaxy phone/tablet lack some features compared to other Samsung Galaxy models? The formula above is used to find the value of the Gamma function for any real value of z. Lets say you want to calculate (4.8). For $x_0 < x < x_1$, take $$e^{-t} \cdot (t^{x_1-1} + t^{x_0-1})\cdot \ln t$$ as the dominating function. $$ -\log(4)\Gamma(2z+1)4^{-z}\Gamma^{-2}(z+1)\}. $\psi(x)=\frac{d}{dx}\log(\Gamma(x))$, http://www.wolframalpha.com/input/?i=integrate+log%28sin%28x%29%29+from+x%3D0+to+x%3Dpi%2F2. In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: [1] [2] It is the first of the polygamma functions. \tag*{} Rearranging this, we have that \displaystyle \Gamma'(z) = \Gamma(z. Python code is used to generate the beautiful plots above. Fisher et al. where $\psi$ is the digamma function. Here's what I've got, using differentiation under the integral. \begin{align} \int_0^{\pi/2}\sin^{2n}(x)\,dx=\frac{2n-1}{2n}\frac{2n-3}{2n-2}\cdots\frac{1}{2}\frac{\pi}{2}=\frac{(2n)! Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity. B(n + 1 2, 1 2): / 2 0 sin2n(x)dx = . The gamma function is applied in exact sciences almost as often as the wellknown factorial symbol . To learn more, see our tips on writing great answers. It is also mentioned there, that when x is a positive integer, k = 1 ( 1 k 1 k + x 1) = k = 1 x 1 1 k = H x 1 where H n is the n th Harmonic Number. Should I give a brutally honest feedback on course evaluations? I didn't even mention it can be defined over the complex numbers as well. \sin^{2z} (x) \ \mathrm{d}x = \frac{\pi}{2}\Gamma(2z+1)4^{-z}\Gamma^{-2}(z+1) MathJax reference. }{2 \Gamma(n+3/2)} Then the above dominates for all y ( x 0, x 1). Derivative of the Gamma Function Unit Aug 21, 2009 Aug 21, 2009 #1 Unit 182 0 A very vague question: What is the derivative of the gamma function? \Gamma'(1) = \int_{0}^{\infty} e^{-t} \, \ln(t) \, dt. \end{align} Setting $x=1$ leads to then differentiating both sides with respect to $z$ gives $$ Hence an analytic continuation of $\int_0^{\pi/2}\sin^{2n}(x)\,dx $ is Why is the eastern United States green if the wind moves from west to east? Because we want to generalize the factorial! $$ You look at some specific x. $$\Gamma(z+1)=e^{-\gamma z}\cdot\prod_{n\geq 1}\left(1+\frac{z}{n}\right)^{-1}e^{z/n}\tag{1}$$ $$ Can virent/viret mean "green" in an adjectival sense? Should I give a brutally honest feedback on course evaluations? If he had met some scary fish, he would immediately return to the surface. Why doesn't the magnetic field polarize when polarizing light. The proof arises from expressing the Gamma Function in the Weierstrass Form, taking a natural logarithm of both sides and then differentiating. How can I fix it? The gamma function then is defined as the analytic continuationof this integral function to a meromorphic functionthat is holomorphicin the whole complex plane except zero and the negative integers, where the function has simple poles. = 1 * 2 * * x, cannot be used directly for fractional values because it is only valid when x is a whole number. What happens if you score more than 99 points in volleyball? special-functions gamma-function. &=& -\frac{\pi}{2}\log(4)=-\pi\log(2). Should teachers encourage good students to help weaker ones? Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. -2\Gamma(2z+1)4^{-z}\Gamma^{-3}(z+1)\psi(z+1) \right. Regarding the two expressions and your doubt about their equality: The equality of $\displaystyle \frac{\Gamma(2z+1)}{4^z\Gamma^2(z+1)}\frac{\pi}{2}$ and $\displaystyle \frac{\Gamma(z+\frac{1}{2})}{\Gamma(z+1)}\frac{\sqrt\pi}{2}$ can be shown by using the fact that $\Gamma(z)\Gamma(z+\frac12)=2^{1-2z}\sqrt{\pi}\Gamma(2z)$ (see wiki): $$\Gamma(2z+1)=\Gamma\left(2\left(z+\frac12\right)\right) = \frac{\Gamma\left(z+\frac12\right)\Gamma\left(z+1\right)}{2^{-2z}\sqrt{\pi}}$$, Thus, We conclude that where the quantitiy $\pi/2$ results from the fact that Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. $$\frac{\Gamma(2z+1)}{4^z\Gamma^2(z+1)}\frac{\pi}{2}=\frac{\Gamma\left(z+\frac12\right)\Gamma\left(z+1\right)}{2^{-2z}\sqrt{\pi}4^z\Gamma^2(z+1)}\frac{\pi}2=\frac{\Gamma(z+\frac{1}{2})}{\Gamma(z+1)}\frac{\sqrt\pi}{2}$$, An excellent discussion of this topic can be found in the book The Gamma Function by James Bonnar. How is this done? \int_0^{\pi/2}\sin^{2n}(x)\,dx=\frac{2n-1}{2n}\frac{2n-3}{2n-2}\cdots\frac{1}{2}\frac{\pi}{2}=\frac{(2n)! \Gamma'(1)=-\gamma, Hence an analytic continuation of $\int_0^{\pi/2}\sin^{2n}(x)\,dx $ is $$ Given a point of the manifold , a vector field : defined in a neighborhood of p . But how to bound $f_h(t)=e^{-t} t^{x-1} \frac{t^h-1}{h}$ by a $L^1(0,\infty)$ function? 2\int_0^{\pi/2}\log(\sin(x))\,dx&=&\frac{\pi}{2}(-2\gamma+2\gamma-\log(4))\\ The following functions are available in R: gamma to compute gamma function; digamma to compute derivative of log gamma function; pgamma to compute incomplete gamma function? 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