# Gamma function

Retired since 2013
28 de Sep de 2014
1 de 45

### Gamma function

• 1. 1 Gamma Function SOLO HERMELIN Updated 28.10.12http://www.solohermelin.com
• 2. 2 SOLO TABLE OF CONTENT Gamma Function Gamma Function History Gamma Function: Euler’s Second Integral Properties of Gamma Function Other Gamma Function Definitions: Gauss’ Formula Some Special Values of Gamma Function: Bohr-Mollerup-Artin Theorem Other Gamma Function Definitions: Weierstrass’ Formula Differentiation of Gamma Function Beta Function: Euler’s First Integral Euler Reflection Formula Duplication and Multiplication Formula Stirling Approximation Formula References
• 3. 3 SOLO Gamma Function History The Gamma Function was first introduced by the Swiss mathematician Leonhard Euler (1707 – 1783). His goal was to generalize the factorial to non-integer values. Later, it was studied by Adrien-Marie Legendre (1752-1833), Carl Friedrich Gauss (1777-1855), Christoph Gudermann (1798-1852), Joseph Liouville (1809 – 1882), Karl Weierstrass (1815- 1897), Charles Hermite (1822-1901),…and others Leonhard Euler )1707–1783( ( ) ( ) 0ln 1 0 1 >−=Γ ∫ = = − xtdtz t t x Adrien-Marie Legendre )1752–1833( The problem of extending the factorial to non-integer arguments was apparently first considered by Daniel Bernoulli and Christian Goldbach in the 1720s, and was solved at the end of the same decade by Leonhard Euler. Euler gave two different definitions: the first was not his integral but an infinite product, ∏ ∞ = +       + = 1 1 1 1 ! k n k n k n of which he informed Goldbach in a letter dated October 13, 1729. He wrote to Goldbach again on January 8, 1730, to announce his discovery of the integral representation Gamma Function
• 4. 4 SOLO Gamma Function History Leonhard Euler )1707–1783( ( ) ( ) 0ln 1 0 1 >−=Γ ∫ = = − xtdtz t t x During the years 1729 and 1730, Euler introduced the following analytic function, By changing of variables we can obtain more known forms ( ) ( ) ( ) 0ln 0 10 1 1 0 1 >=−=−=Γ ∫∫∫ ∞= = −= ∞= −− = −= = = − − − xtd e t tdetuduz t t t xt t tx eu dtedu u u x t t ( ) ( ) ( ) ( ) 022ln 0 12 0 12 2 1 0 1 22 2 2 >=−=−=Γ ∫∫∫ ∞= = −− = ∞= −− = −= = = − − − xtdettdettuduz t t tx t t tx eu dtetdu u u x t t The notation Γ (x) is due to Legendre in 1809, while Gauss used Π (x) = Γ (x+1) Carl Friedrich Gauss (1777 – 1855) Adrien-Marie Legendre )1752–1833( Gamma Function
• 5. 5 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Proof: Gamma Function 0& >+= xyixz ∫∫∫ ∞= = −= += −∞= += − += t t t zt t t zt t t z td e t td e t td e t 1 11 0 1 0 1 For the first part: x t xx t x tdttd e t td e t x t t t x t t x et t t yixt t t z t 1 lim 111 0 1 0 1 0 1 11 0 11 0 1 =−==≤= +→ = += = += − >= += −+= += − ∫∫∫ The first integral converges for any x ≥ δ > 0. For the second integral, using integration by parts: ( ) ( ) ( ) ( ) ( )( ) ∫ ∫∫∫∫ ∞= = − ∞= = −− = = ∞= = − ∞= = −− = = ∞= = −∞= = −+∞= = − −−+−−+= −+−=== − − − − t t t x e t t tx edv tu t t t x e t t tx edv tu t t t xt t t yixt t t z td e t xxetx e td e t xettd e t td e t td e t t x t x 1 3 /1 1 2 1 2 /1 1 1 1 1 1 1 1 1 211 1 1 2 1   Euler’s Second Integral Gamma integral is defined, and converges uniformly for x > 0. Gamma Function
• 6. 6 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Proof (continue): Gamma Function 0& >+= xyixz For the second integral, using integration by parts: ( ) ( ) ( ) ( ) ( )( ) ∫ ∫∫∫∫ ∞= = − ∞= = −− = = ∞= = − ∞= = −− = = ∞= = −∞= = −+∞= = − −−+−−+= −+−=== − − − − t t t x e t t tx edv tu t t t x e t t tx edv tu t t t xt t t yixt t t z td e t xxetx e td e t xettd e t td e t td e t t x t x 1 3 /1 1 2 1 2 /1 1 1 1 1 1 1 1 1 211 1 1 2 1   After [x] (the integer defined such that x-[x] < 1) such integration the power of t in the integrand becomes x-[x]-1 < 0. and we have: ( )( ) [ ]( ) [ ]( ) ( )( ) [ ]( ) ∞<−−−<−−− ∫∫ ∞= = ∞= = −− t t t t t txx td e xxxxtd et xxxx 11 1 1 21 1 21  Therefore the Gamma integral is defined, and converges uniformly for x > 0. Gamma integral is defined, and converges uniformly for x > 0. q.e.d. Gamma Function Return to Table of Content
• 7. 7 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Proof : Gamma Function 0& >+= xyixz ( ) ( )zzz Γ=+Γ 1 ( )  ( ) ( ) ( )zztdetztdtzeettdetz t t tz t t ud z v t v t u z dtedvtu partsby t t tz tz Γ=+=−−−==+Γ ∫∫∫ ∞= = −− ∞= = −− ∞ − ==∞= = − − 0 1 0 1 0 , nintegratio 0 01  Properties of Gamma Function : 1 Note that for the evaluation of Gamma Function for a Positive Real Number we need to know only the value of Γ (x) for 0 < x < 1 ( ) ( ) ( ) ( ) ( )xxxnxnxnx Γ+−+−+=+Γ 121  ( ) ( ) ( ) ( ) ( )121 −+−++ +Γ =Γ nxnxxx nx x  For x < 0 with –n < x < -n+1 or 0 < x+n < 1, we define We can see that for x = 0 or a negative integer the denominator of the right side is zero, and so Γ (x) is undefined (goes to infinity) Gamma Function ( ) ,2,1,0!1 ==+Γ nnn
• 8. 8 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Proof : Gamma Function ( ) ( ) ( )!1 1 Residue 1 1 − − =Γ − +−→ n z n nz Residues of Gamma Function at x = 0,-1, -2,---,-n,..: ( ) ( ) ( ) ( ) ( )121 −+−++ +Γ =Γ nxnxxx nx x  q.e.d. ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( )( ) ( ) ( ) ( )!1 1 121 1 121 1limResidue 1 1 11 − − = −+−+− Γ = −+−++ +Γ −+=Γ − +−→+−→ nnn nxnxxx nx nxx n nxnx   Gamma Function
• 9. 9 SOLO Gamma Function Γ (x) and its Inverse 1/Γ (x) Gamma Function
• 10. 10 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Absolute value |Γ (z)| Real value ReΓ (z) Imaginary value ImΓ (z) Gamma Function
• 11. 11 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Absolute value |Γ (z)| Gamma Function
• 12. 12 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function ( ) ( )zzz Γ=+Γ 1 Let compute ( ) 11 0 0 =−==Γ ∞− ∞= = − ∫ t t t t etde Therefore for any n positive integer: ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )!1122112111 −=Γ−−=−Γ−−=−Γ−=Γ nnnnnnnnn  Properties of Gamma Function : 1 2 q.e.d. Gamma Function
• 13. 13 SOLO Primes Second definition identical to First ( )[ ] ( ) ( ) ( ) ( ) ( )bayxallyfxfyxf ,,1,011 ∈∈−+≤−+ λλλλλ Convex Function : A Function f (x) is called Convex in an interval (a,b) if for every x,y (a,b) we haveϵ A Function f (x), defined for x > 0, is called Convex, if the corresponding function ( ) ( ) ( ) y xfyxf y −+ =φ defined for all y > -x, y ≠ 0, is monotonic Increasing throughout the range of definition. If 0 < x1 < x < x2, are given by choosing y1 = x1 – x < 0, y2 = x2 – x > 0, we express the condition of convexity as ( ) ( ) ( ) ( ) ( ) ( ) xx xfxf y xx xfxf y − − =≤ − − = 2 2 2 1 1 1 φφ ( ) ( )[ ] ( ) ( ) ( )[ ] ( )xxxfxfxxxfxf −−≥−− 1221 ( ) ( ) ( ) ( ) ( ) ( ) ( ) λλ − − − + − − ≤ 1 12 1 2 12 2 1 xx xx xf xx xx xfxf One other equivalent definition:
• 14. 14 SOLO Primes ( )[ ] ( ) ( ) ( ) ( )1,0ln1ln1ln ∈−+≤−+ λλλλλ yfxfyxf Logarithmic Convex Function : A Function f (x)>0 is called logarithmic-convex or simply log-convex if ln (f (x) ) is convex or This is equivalent to ( )[ ] ( ) ( )( )λλ λλ − ≤−+ 1 ln1ln yfxfyxf Since the logarithm is a momotonic increasing function we obtain ( )[ ] ( ) ( )( ) ( ) yxyfxfyxf <∈≤−+ − ,1,01 1 λλλ λλ
• 15. 15 SOLO Primes ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Proof : Gamma Function 0& >+= xyixz ( )[ ] ( ) ( ) ( ) ( )1,0ln1ln1ln ∈Γ−+Γ≤−+Γ λλλλλ baba Properties of Gamma Function : 3 Gamma is a Log Convex Function ( )[ ] ( ) ( ) ( ) ( ) ( ) λλ λλ λλλλ λλ − −∞ −− ∞ −− ∞ −−−−− ∞ −−−+ ΓΓ=                ≤ ==−+Γ ∫∫ ∫∫ 1 1 0 1 0 1 0 111 0 11 1 badtetdtet dtetetdtetba tbta InequalityHolder tbtatba q.e.d. Return to Table of Content
• 16. 16 SOLO Primes ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Proof : Gamma Function Other Gamma Function Definitions: ( ) ( ) ( )nxxx nn x x n ++ =Γ ∞→ 1 ! limGauss’ Formula Since the Gamma Function is monotonically increasing the logarithm of Gamma Function is also monotonic increasing and for 0 < x < 1 and any n > 2 we have ( ) ( ) ( ) nnx nnx −+ Γ−+Γ lnln ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ] [ ] ( )[ ] ( )            −      − − −− ≤ −−+Γ ≤ − −−− !1 ! ln !2 !1 ln 1 !1ln!ln!1lnln 1 !1ln!2ln n n n n nn x nnxnn ( ) ( ) ( ) n x n nx n ln !1 ln 1ln ≤ − +Γ ≤− ( ) ( ) ( ) 1 1 ln1ln −=← ≤ −+− Γ−+−Γ x nn nn ( ) ( ) ( ) nn nnx −+ Γ−+Γ ≤ →= 1 ln1ln1 Carl Friedrich Gauss (1777 – 1855)
• 17. 17 SOLO Primes ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Proof (continue - 1) : Gamma Function Other Gamma Function Definitions: Since the Gamma Function is monotonically increasing the logarithm of Gamma Function is also monotonic increasing and for 0 < x < 1 and any n > 2 we have ( ) ( ) ( ) n x n nx n ln !1 ln 1ln ≤ − +Γ ≤− ( ) ( ) ( ) xx n n nx n ln !1 ln1ln ≤ − +Γ ≤− 10 << x ( ) ( ) ( ) ( )!1!11 −≤+Γ≤−− nnnxnn xx Use ( ) ( ) ( ) ( ) ( )xxxnxnxnx Γ+−+−+=+Γ >     0 121 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) xxnxnx nn x xxnxnx nn xx 121 !1 121 !11 +−+−+ − ≤Γ≤ +−+−+ −−  ( ) ( ) ( )nxxx nn x x n ++ =Γ ∞→ 1 ! limGauss’ Formula Euler 1729 Gauss 1811
• 18. 18 SOLO Primes ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Proof (continue - 2) : Gamma Function Other Gamma Function Definitions: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) xxnxnx nn x xxnxnx nn xx 121 !1 121 !11 +−+−+ − ≤Γ≤ +−+−+ −−  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) xxnxnx nn x xxnxnx nn xx 11 !1 11 ! +−++ + ≤Γ≤ +−++  Take the limit n → ∞ ( ) ( ) ( ) ( ) ( ) ( ) ( ) xxnxnx nn n x xxnxnx nn x n x n x n 11 ! lim 1 1lim 11 ! lim 1 +−++       +≤Γ≤ +−++ ∞→∞→∞→    ( ) ( ) ( ) ( ) ( )1,0 11 ! lim ∈ +−++ =Γ ∞→ x xxnxnx nn x x n  Substitute n+1 for n ( ) ( ) ( )nxxx nn x x n ++ =Γ ∞→ 1 ! limGauss’ Formula
• 19. 19 SOLO Primes ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Let substitute x + 1 for x Gamma Function Other Gamma Function Definitions: ( ) ( ) ( ) ( ) ( ) ( )1,0 11 ! lim ∈ +−++ =Γ Γ ∞→ x xxnxnx nn x x x n n     q.e.d ( ) ( ) ( )nxxx nn x x n ++ =Γ ∞→ 1 ! limGauss’ Formula Proof (continue - 3) : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1,0 11 ! lim 1 lim 11 ! lim1 1 1 ∈Γ= +−++++ = ++++ =+Γ Γ ∞→∞→ + ∞→ xxx xxnxnx nn nx n x xnxnx nn x x x nn x n         The right side is defined for 0 < x <1. The left side extend the definition for (1 , 2). Therefore the result is true for all x , but 0 and negative integers. Return to Table of Content
• 20. 20 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Other Gamma Function Definitios: Start from Gauss Formula ( ) ( )xx n n Γ=Γ ∞→ lim q.e.d ( ) constantMascheroni-Euler57721566.0ln 1 2 1 1lim 11 ≈      −+++= + =Γ ∞→ ∞ = − ∏ n n k x e x e x n k k x x γ γ Weierstrass’ Formula Proof : ( ) ( ) ( ) ( )       +      − +      + =       +      − +      + = +−++ =Γ       −−−− n x n xx x eee e x x n x n x n xxnxnx nn x n xxx n nx xx n 1 1 1 1 1 1 1 1 11 11 ! : 21 1 2 1 1ln      ( ) ( ) ∏∏ ∞ = − =       −−−− ∞→∞→ + = + =Γ=Γ 11 1 2 1 1ln 11 1 limlim k k x xn k k x n nx n n n k x e x e k x e x exx γ Karl Theodor Wilhelm Weierstrass (1815 – 11897) Gamma Function Return to Table of Content
• 21. 21 SOLO Primes ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Some Special Values of Gamma Function: q.e.d ( ) π π ====Γ ∫∫ ∞= = − = = ∞= = − 2 222/1 0 2 0 2 2 t t u ut duudt t t t udetd t e ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) πn n nnnnn 2 12531 2/12/112/32/12/12/12/1 −⋅⋅ =Γ+−−=−Γ−=+Γ   ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) π 12531 21 2/12/32/1 2/1 2/1 2/3 2/1 −⋅⋅ − = −+−+− Γ = +− +−Γ =+−Γ nnnn n n nn  ( ) π=Γ 2/1 ( ) ( ) πn n n 2 12531 2/1 −⋅⋅ =+Γ  ( ) ( ) ( ) π 12531 21 2/1 −⋅⋅ − =+−Γ n n nn  Proof: Return to Table of Content
• 22. 22 SOLO Harald August Bohr ( 1887 – 1951) Proof: Choose n > 2, and 0 < x < 1 and let 11 +≤+<<− nxnnn By logarithmic convexity of f (x), we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) nn nfnf nxn nfxnf nn nfnf −+ −+ ≤ −+ −+ ≤ −− −− 1 ln1lnlnln 1 ln1ln ( ) ( ) ( ) ( ) ( ) 1 !1ln!ln!1lnln 1 !1ln!2ln −− ≤ −−+ ≤ − −−− nn x nxnfnn By the second property ( ) ( ) ( ) ( ) ( ) !1,!1,!21 nnfnnfnnf =+−=−=− ( ) ( )( ) ( ) ( )xfxxxnxnxnf 121 +−+−+=+  ( ) ( ) ( ) xx n n xnf n ln !1 ln1ln ≤ − + ≤− Emil Artin (1898 – 1962) Hamburg University Johannes Mollerup (1872 – 1937) Gamma Function Bohr-Mollerup-Artin Theorem: The theorem characterizes the Gamma Function, defined for x > 0 by as the only function f (x) on the interval x > 0 that simultaneously has the three properties • f (1) = 1 • f (1+x) = x f (x) for x > 0 • f is logarithmically convex or Gauss Formula( ) ∫ ∞= = −− =Γ t t tz tdetz 0 1 ( ) ( ) ( )nxxx nn z x n ++ =Γ ∞→ 1 ! lim
• 23. 23 SOLO Bohr-Mollerup-Artin Theorem: Harald August Bohr ( 1887 – 1951) The theorem characterizes the Gamma Function, defined for x > 0 by as the only function f (x) on the interval x > 0 that simultaneously has the three properties • f (1) = 1 • f (1+x) = x f (x) for x > 0 • f is logarithmically convex Proof (continue-1): By the second property ( ) ( )( ) ( ) ( )xfxxxnxnxnf 121 +−+−+=+  ( ) ( )( ) ( ) ( ) ( ) xx n n xfxxxnxn n ln !1 121 ln1ln ≤ − +−+−+ ≤−  We found Since lan is a monotonic increasing function, we have ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )121 !1 121 !11 −+−++ − ≤≤ −+−++ −− xnxnxx nn xf xnxnxx nn xx  ( ) ( )( ) ( ) ( ) ( )( ) ( ) x xxx n n xnxnxx nn xf xnxnxx nn 1 11 ! 11 ! + +−++ ≤≤ +−++  n n ↓ −1 ( ) ∫ ∞= = −− =Γ t t tz tdetz 0 1 ( ) ( ) ( )nxxx nn x x n ++ =Γ ∞→ 1 ! limor Gauss Formula Emil Artin (1898 – 1962) Hamburg University Johannes Mollerup (1872 – 1937) Gamma Function
• 24. 24 SOLO Bohr-Mollerup-Artin Theorem: q.e.d. Harald August Bohr ( 1887 – 1951) The theorem characterizes the Gamma Function, defined for x > 0 by as the only function f (x) on the interval x > 0 that simultaneously has the three properties • f (1) = 1 • f (1+x) = x f (x) for x > 0 • f is logarithmically convex Johannes Mollerup (1872 – 1937) Proof (continue - 2): ( ) ( )( ) ( ) ( ) ( )( ) xxx nxnxnxx nn xf xnxnxx nn       + +−++ ≤≤ +−++ 1 1 11 ! 11 !  ( ) ∫ ∞= = −− =Γ t t tz tdetz 0 1 ( ) ( ) ( )nxxx nn x x n ++ =Γ ∞→ 1 ! limor Gauss Formula By taking n → ∞ we obtain ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )         1 1 1lim 11 ! lim 11 ! lim x n x x n x x n nxnxnxx nn xf xnxnxx nn       + +−++ ≤≤ +−++ ∞→ Γ ∞→ Γ ∞→ But this is possible only if ( ) ( )xxf Γ= Emil Artin (1898 – 1962) Hamburg University Gamma Function
• 25. 25 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Gamma integral is defined, and converges uniformly for x > 0. Differentiation of Gamma Function: q.e.d ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,2 !11' ln 0 1''' ln constantMascheroni-Euler57721566.0 111' ln 1 1 1 1 22 2 2 2 1 >≥ + −− = Γ Γ =Γ > + = Γ Γ−ΓΓ =Γ ≈      + −+−−= Γ Γ =Γ ∑ ∑ ∑ ∞ = − − ∞ = ∞ = xn kx n x x xd d x xd d kxx xxx x xd d kxkxx x x xd d k n n n n n n k k γγ Proof : Start from Weierstrass Formula ( ) ∏ ∞ = − + =Γ 1 1k k x x k x e x e x γ ( ) ∑∑ ∞ = ∞ =       +−+−−=Γ 11 1lnlnln kk k x k x xxx γ ( ) ∑∑ ∞ = ∞ = + −+−−=Γ 11 1 1 11 ln kk k x k kx x xd d γ ( ) ( ) ( ) 0 111111 ln 0 2 1 22 1 2 2 > + = + +=            + −+−−=Γ ∑∑∑ ∞ = ∞ = ∞ = kkk kxkxxkxkxxd d x xd d γ ( ) ( ) ( ) ( ) ( ) ( )∑ ∞ = − − + −− = Γ Γ =Γ 0 1 1 !11' ln k n n n n n n kx n x x xd d x xd d Gamma Function We can see that ( ) ( ) ( ) γγ −=      + −+−−= Γ Γ ==Γ + − = ∞→ ∑    1 1 1 1 1 11 lim 1 1 1 1' 1ln n n k n kk x xd d Return to Table of Content
• 26. 26 SOLO ( ) ( )∫ = = −− −= 1 0 11 1, s s zy sdsszyBBeta Function Beta Function is related to Gamma Function: ( ) ∫∫ ∞= = −− = =∞= = −− ==Γ u u uy duudt utt t ty udeutdety 0 12 2 0 1 2 2 2 ( ) ( ) ( ) ( )zy zy zyB +Γ ΓΓ =, Proof: In the same way: ( ) ∫ ∞= = −− =Γ v v vz vdevz 0 12 2 2 ( ) ( ) ( ) ∫ ∫ ∞= = ∞= = +−−− =ΓΓ u u v v vuuzy vdudevuzy 0 0 1212 22 4 Use polar coordinates: ϕϕ ϕϕ ϕϕ ϕ ϕ ϕ ϕ ϕ drdrdrd r r drd vrv uru vdud rv ru = − = ∂∂∂∂ ∂∂∂∂ =    = = cossin sincos // // sin cos ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                     = =ΓΓ ∫∫ ∫ ∫ = = −− +Γ ∞= = −−+ ∞= = = = −−−−+ 2/ 0 1212 0 12 0 2/ 0 121212 sincos22 sincos4 2 2 πϕ ϕ πϕ ϕ ϕϕϕ ϕϕϕ drder drderzy zy zy r r rzy r r rzyzy    Euler’s First Integral Gamma Function
• 27. 27 SOLO ( ) ( )∫ = = −− −= 1 0 11 1, s s zy sdsszyBBeta Function Euler’s First Integral Beta Function is related to Gamma Function: ( ) ( ) ( ) ( )zy zy zyB +Γ ΓΓ =, Proof (continue): ( ) ( ) ( ) ( ) ( )         +Γ=ΓΓ ∫ = = −− 2/ 0 1212 sincos2 πϕ ϕ ϕϕϕ dzyzy zy Change variables in the integral using ϕϕϕϕ dsds cossin2sin2 == ( ) ( ) ( ) ( )zyBsdssd s s yzzy ,1sincos2 1 0 11 2/ 0 1212 =−= ∫∫ = = −− = = −− πϕ ϕ ϕϕϕ ( ) ( ) ( ) ( )zyBzyzy ,+Γ=ΓΓTherefore q.e.d. Use z→y and y → 1 - z ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫ ∫ ∞= = −∞= = − − −+ = + = = = −− + = +       + − + = −=−Γ=−ΓΓ u u zu u z z zu u s u ud sd s s zz ud u u u ud u u u u dssszzBzz 0 1 0 21 11 1 1 0 1 111 1 1 11,11 2 q.e.d. Gamma Function Return to Table of Content
• 28. 28 SOLO Proof ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )yzBzyzyBzyyz yzBzyB ,, ,, +Γ=+Γ=ΓΓ = Use y → 1 - z ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫ ∫ ∞= = −∞= = − − −+ = + = = = −− + = +       + − + = −=−Γ=−ΓΓ u u zu u z z zu u s u ud sd s s zz ud u u u ud u u u u dssszzBzz 0 1 0 21 11 1 1 0 1 111 1 1 11,11 2 ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Other Gamma Function Properties: ( ) ( ) ( )z zz π π sin 1 =−ΓΓ Euler Reflection Formula Gamma Function
• 29. 29 SOLO Proof (continue - 1) ( ) ( ) ∫ ∞= = − + =−ΓΓ u u x ud u u xx 0 1 1 1 ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Other Gamma Function Properties: Replace the path from 0 to ∞ by the Hankel contour Hε in the Figure, described by four paths, traveled in counterclockwise direction: 1. going counterclockwise above the real axis, (u = |u|) 2. along the circular path CR, 3. bellow the real axis, (u= |u|e -2πi ) 4. along the circular path Cε. ∫∫∫∫ + − + − + + + −− − −− εε π ε C yR y yi C yR y ud u u ud u u eud u u ud u u R 1111 2 Define y = 1 – x, and assume x,y (0,1)ϵ ( ) ( ) ( )z zz π π sin 1 =−ΓΓ Euler Reflection Formula Gamma Function
• 30. 30 SOLO Proof (continue - 1) ( ) ( ) ∫ ∞= = − + =−ΓΓ u u x ud u u xx 0 1 1 1 ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Other Gamma Function Properties: This path encloses the pole u=-1 of that has the residue 1+ − u u y yi eu y y eu u u i π π − =−= − − ==      + 11 Residue By the Residue Theorem For z ≠ 0 we have ( ) yzyzyzyy zeeez −−−−− ==== lnlnReln ( ) ( ) ( )z zz π π sin 1 =−ΓΓ Euler Reflection Formula ( ) yi y eu y C yR y iy C yR y ei u u ui u u izd z z ud u u ezd z z ud u u i R π ε π ε ππ π π ε − − =−→ −−− − −− =            + +=       + = + − + − + + + − ∑∫∫∫∫ 2 1 1lim2 1 Residue2 1111 1 2 Gamma Function
• 31. 31 SOLO Proof (continue - 2) ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Other Gamma Function Properties: yi C yR y iy C yR y eizd z z ud u u ezd z z ud u u R π ε π ε π ε − −− − −− = + − + − + + + ∫∫∫∫ 2 1111 2 For the second and forth integral we have ( ) 0 lnlnReln ≠==== −−−−− zzeeez yzyzyzyy z z z z z z yyy − ≤ + ≤ + −−− 111 Hence for small ε we have: and for large R we have: 0 1 2 1 01 →−− → − ≤ +∫ ε ε ε π ε y C y zd z z 0 1 2 1 1 ∞→−− → − ≤ +∫ Ry C y R R zd z z R π Therefore the integrals on the circular paths are zero for ε→0 and R →∞ ( ) ( ) ( )z zz π π sin 1 =−ΓΓ Euler Reflection Formula Gamma Function
• 32. 32 SOLO Proof (continue - 3) ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Other Gamma Function Properties: yi y iy y eiud u u eud u u ππ π − ∞ − − ∞ − = + − + ∫∫ 2 11 0 2 0 We obtain Multiply both sides by yi e π+ ( ) iud u u ee y iyiy πππ 2 10 = + − ∫ ∞ − − ( ) ( )yee i ud u u iyiy y π π π ππ sin 2 10 = − = + − ∞ − ∫Rearranging we obtain Since both sides of this equation are meromorphic (analytic) in x (0,1) we canϵ extend the result for all analytic parts of z C (complex plane).ϵ ( ) ( ) ( )[ ] ( ) ( )1,0 sin1sin11 1 0 1 0 1 ∈= − = + = + =−ΓΓ ∫∫ ∞= = −−=∞= = − x xx ud u u ud u u xx u u yxyu u x π π π π Substituting y = 1 – x we obtain ( ) ( ) ( )z zz π π sin 1 =−ΓΓ Euler Reflection Formula Gamma Function
• 33. 33 SOLO Onother Proof ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Other Gamma Function Properties: Start with Weierstrass Gamma Formula ( ) ( ) ( )z zz π π sin 1 =−ΓΓ Euler Reflection Formula ( ) ∏ ∞ = − + =Γ 1 1k k x x k x e x e x γ ( ) ( ) ∏∏ ∞ = ∞ = − −       −−= −+ −= −ΓΓ 1 2 2 2 1 2 1 11 1 kk k x k x xx k x x e k x e k x eex xx γγ Use the fact that Γ (-x)=- Γ (1-x)/x to obtain ( ) ( ) ∏ ∞ =       −= −ΓΓ 1 2 2 1 1 1 k k x x xx Now use the well-known infinite product ( ) ∏ ∞ =       −= 1 2 2 1sin k k x xx ππ q.e.d. Gamma Function
• 34. 34 SOLO Proof ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Other Gamma Function Properties: ( )z zz π π cos2 1 2 1 =      −Γ      +Γ Start from Substitute ½ +z instead of z ( ) ( ) ( )z zz π π sin 1 =−ΓΓ ( )z z zz π π π π cos 2 1 sin 2 1 2 1 =             + =      −Γ      +Γ q.e.d. Gamma Function Return to Table of Content
• 35. 35 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Stirling Approximation Formula: ( ) 121 >>≈+Γ − xexxx xx π ( ) ( ) ( ) ( ) ( ) ( )( ) ∫ ∫∫∫ ∞= −= ++−−+ ∞= −= −−+ ∞= −= +− += = ∞= −= − = +=+==+Γ u u uuxxx u u xuxxx u u xxux uxt udxtd t t xt udeex udueexudxuxetdtex 1 1ln1 1 1 1 1 1 1 111 Proof: The function f(u) = -u + ln (1 + u) equals zero for u = 0. For other values of u we have f(u) < 0. This implies that the integrand of the last integral equals 1 at u = 0 and that this integrand becomes very small for large values of x at other values of u. So for large values of x we only have to deal with the integrand near u = 0. Note that we have ( ) ( ) ( ) ( ) 0 2 1 2 1 1ln 2222 →Ο+−=Ο+−+−=++−= uforuuuuuuuuuf This implies that ( )( ) ∞→≈ ∫∫ ∞= −∞= − ∞= −= ++− xforduedue u u ux u u uux 2/ 1 1ln 2 James Stirling, a contemporary of Euler, also attempted to find a continuous expression for the factorial and came up with what is now known as Stirling's formula. Although Stirling's formula gives a good estimate of , also for non-integers, it does not provide the exact value. Extensions of his formula that correct the error were given by Stirling himself and by Jacques Philippe Marie Binet‫ן‬ Gamma Function
• 36. 36 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Stirling Approximation Formula: ( ) 121 >>≈+Γ − xexxx xx π Proof (continue): ( ) ( )( ) ∞→≈=+Γ ∫∫ ∞= −∞= −−+ ∞= −= ++−−+ xfordueexudeexx u u uxxx u u uuxxx 2/1 1 1ln1 2 1 ∞→== − ∞= −∞= −− = = ∞= −∞= − ∫∫ xforxdtexdue t t t xtu xtdud u u ux π π 22 2/12/1 /2 /2 2/ 22  If we set we have by using the normal integralxtu /2= therefore: ( ) ∞→≈+Γ − xexxx xx π21 q.e.d. Gamma Function
• 37. 37 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Duplication and Multiplication Formula: ( ) ( ) 0Re2 22 1 12 >Γ=      +ΓΓ − zzzz z π Legendre Duplication Formula 1809 Adrien-Marie Legendre )1752–1833( Proof: ( ) ( ) ( ) ( ) ( ) ( ) ( )2/1,2sin22sin2 2sin22sincos2, 21 2/ 0 1221 0 1221 2/ 0 1221 2/ 0 1212 zBdd ddzzB zzzzz zzzz ⋅=⋅⋅== ⋅== −−−−− −−−− ∫∫ ∫∫ ππ ππ ττττ ϕϕϕϕϕ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0Re 2/1 2/1 22/1,2, 2 2121 > +Γ Γ⋅Γ ⋅=⋅== Γ Γ⋅Γ −− z z z zBzzB z zz zz We have therefore q.e.d( )  ( ) 0Re2 22 1 12 2 1 >Γ=      +ΓΓ −       Γ zzzz z π Gamma Function
• 38. 38 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Duplication and Multiplication Formula: ( ) ( )( ) ( )znn n n z n z n zz znn Γ=      − +Γ      +Γ      +ΓΓ −− 2/12/1 2 121 π Gauss Multiplication Formula Proof: n z 1 = Carl Friedrich Gauss (1777 – 1855)( )( ) nn n nn n 2/1 2121 − =      − Γ      Γ      Γ π  Euler Multiplication Formula Gamma Function Define the function: ( )       −+ Γ      + Γ      Γ= n nx n x n x nxf x 11 :  This function has the following properties: 1 ( ) ( )xfx n x n x n nx n x n x nn n nx n nx n x n x nxf x x ⋅=      Γ⋅⋅      −+ Γ      + Γ      + Γ⋅=       + Γ      −+ Γ      + Γ      + Γ=+ ↓ + 121 121 1 1   
• 39. 39 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Duplication and Multiplication Formula: ( ) ( )( ) ( )znn n n z n z n zz znn Γ=      − +Γ      +Γ      +ΓΓ −− 2/12/1 2 121 π Gauss Multiplication Formula Proof (continue – 1): Carl Friedrich Gauss (1777 – 1855) Gamma Function Since (ln nx )”=(x ln n)”=(ln n)’=0, and each Γ ((x+k)/k) is log convex. f (x) is log convex. ( ) ( )       Γ      Γ⋅      Γ==Γ= n n nn naaf nn  21 11 So using Bohr-Mollerup-Artin Theorem we can write: f (x) = an Γ(x) where an is a constant, to be found, and Γ (1)=1 (the third condition of the Theorem). 2 Therefore Use Gauss’ Formula for Gamma Function with x=k/n ( ) ( )pnknkk npp p n k n k n k pp n k pn k p n k p ++ =       +      + =      Γ + ∞→∞→   1 ! lim 1 ! lim
• 40. 40 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Duplication and Multiplication Formula: ( ) ( )( ) ( )znn n n z n z n zz znn Γ=      − +Γ      +Γ      +ΓΓ −− 2/12/1 2 121 π Gauss Multiplication Formula Proof (continue – 2): Carl Friedrich Gauss (1777 – 1855) Gamma Function ( ) ( )pnknkk npp n k pn k p ++ =      Γ + ∞→  1 ! lim Since k = 1,2,…,p ( ) ( ) [ ] ( ) ( )[ ] ( ) ( )[ ] ( ) ( )( )!1! 11211 nppnn pnnpnnnnnpnknkk p k ⋅+=⋅+= ⋅+⋅+++⋅⋅=⋅++∏=  ( ) ( ) ( ) ( ) ( ) ( )! ! lim ! ! lim 21 2 1 1 1 1 pnn pnp n pnn pnp n n n nn na n pnn p n n npnn p n ⋅+ = ⋅+ =      Γ      Γ⋅      Γ= + + ∞→ ++ + ∞→  
• 41. 41 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Duplication and Multiplication Formula: ( ) ( )( ) ( )znn n n z n z n zz znn Γ=      − +Γ      +Γ      +ΓΓ −− 2/12/1 2 121 π Gauss Multiplication Formula Proof (continue – 3): Carl Friedrich Gauss (1777 – 1855) Gamma Function ( ) ( ) ( )! ! lim 2 1 1 pnn pnp na n pnn p n ⋅+ = + + ∞→ Use the identity ( ) ( ) ( )npp pnpn pnn pn n pnpn ⋅ ⋅ ⋅ ⋅+ =      ⋅ +      ⋅ +⋅      ⋅ += ∞→∞→ 1 ! ! lim1 2 1 1 1lim1  to an to get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 12 1 1 ! ! lim ! ! ! ! lim1 ! ! lim − ⋅ ∞→ + + ∞→ + + ∞→ ⋅ = ⋅⋅ ⋅+ ⋅ ⋅+ =⋅ ⋅+ = n pnn pn n pnn p n pnn p n ppn np n pnpn pnn pnn pnp n pnn pnp na
• 42. 42 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Duplication and Multiplication Formula: ( ) ( )( ) ( )znn n n z n z n zz znn Γ=      − +Γ      +Γ      +ΓΓ −− 2/12/1 2 121 π Gauss Multiplication Formula Proof (continue – 4): Carl Friedrich Gauss (1777 – 1855) Gamma Function to an to get ( ) ( ) 2 1 ! ! lim − ⋅ ∞→ ⋅ = n pnn p n ppn np na ∞→≈ − + pepp p p 2 1 2! π ( ) ( ) ( ) ∞→⋅≈⋅ ⋅−+⋅ pepnpn pnpn 2 1 2! π ( ) ( ) ( ) 2 1 2 1 2 1 2 1 2 1 2 2 2 lim n pepn nep na n n pnpn pn n p p p n − − ⋅−+⋅ ⋅− + ∞→ = ⋅         = π π π Use Stirling’s Approximation formula ( ) ∞→≈+Γ − xexxx xx π21
• 43. 43 SOLO ( ) ∫ ∞= = − =Γ t t t z td e t z 0 1 Gamma Function Duplication and Multiplication Formula: ( ) ( )( ) ( )znn n n z n z n zz znn Γ=      − +Γ      +Γ      +ΓΓ −− 2/12/1 2 121 π Gauss Multiplication Formula Proof (continue – 4): Carl Friedrich Gauss (1777 – 1855) Gamma Function ( ) 2 1 2 1 2 na n n − = π ( ) ( )xa n nx n x n x nxf n x Γ=      −+ Γ      + Γ      Γ= 11 :  We have or ( ) ( )xn n nx n x n x xn Γ=      −+ Γ      + Γ      Γ +−− 2 1 2 1 2 11 π Define x = n z to obtain ( ) ( ) ( )znn n n z n zz znn Γ=      − +Γ      +ΓΓ +−− 2 1 2 1 2 11 π q.e.d Return to Table of Content
• 44. 44 SOLO References Internet http://en.wikipedia.org/wiki/ G.B. Arfken, H.J. Weber, “Mathematical Methods for Physicists”, Academic Press, Fifth Ed., 2001 http://www.frm.utn.edu.ar/analisisdsys/material/function_gamma.pdf http://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf Gamma Function M.Abramowitz & I.E. Stegun, ED., “Handbook of Mathematical Functions”, Dover Publication, 1965, H.Vic Dannon,”Riemann Zeta Function: The Riemann Hypothesis Origin, the Factoriztion Error, and the Count of the Primes”, Gauge Institute Journal, Vol. 5, No. 4, November 2009 J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007 D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010), http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf http://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf Return to Table of Content
• 45. January 6, 2015 45 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

### Notas del editor

1. http://www.frm.utn.edu.ar/analisisdsys/material/function_gamma.pdf
2. http://www.frm.utn.edu.ar/analisisdsys/material/function_gamma.pdf
3. H.Vic Dannon,”Riemann Zeta Function: The Riemann Hypothesis Origin, the FaCTORIZATION Error, and the Count of the Primes”, Gauge Institute Journal, Vol. 5, No. 4, November 2009
4. H.Vic Dannon,”Riemann Zeta Function: The Riemann Hypothesis Origin, the FaCTORIZATION Error, and the Count of the Primes”, Gauge Institute Journal, Vol. 5, No. 4, November 2009
5. M.Abramowitz &amp; I.E. Stegun, ED., “Handbook of Mathematical Functions”, Dover Publication, 1965, pg.255
6. http://en.wikipedia.org/wiki/Gamma_function
7. http://en.wikipedia.org/wiki/Gamma_function
8. J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007 K. Chandrasekharan, “Lectures on The Riemann Zeta-Function”, Tata Institute of Fundamental Research, Bombay, 1953
9. J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007
10. J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007
11. J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007
12. J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007
13. J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007
14. J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007
15. J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007
16. htt[p://en.wikipedia.org/wiki/Bohr_Mollerup_theorem K. Chandrasekharan, “Lectures on The Riemann Zeta-Function”, Tata Institute of Fundamental Research, Bombay, 1953 http://en.wikipedia.org/wiki/Harald_Bohr http://en.wikipedia.org/wiki/Emil_Artin
17. htt[p://en.wikipedia.org/wiki/Bohr_Mollerup_theorem K. Chandrasekharan, “Lectures on The Riemann Zeta-Function”, Tata Institute of Fundamental Research, Bombay, 1953 http://en.wikipedia.org/wiki/Harald_Bohr http://en.wikipedia.org/wiki/Emil_Artin
18. htt[p://en.wikipedia.org/wiki/Bohr_Mollerup_theorem K. Chandrasekharan, “Lectures on The Riemann Zeta-Function”, Tata Institute of Fundamental Research, Bombay, 1953 http://en.wikipedia.org/wiki/Harald_Bohr http://en.wikipedia.org/wiki/Emil_Artin
19. J. Baltzersen, “Hardy’s Theorem and the prime number theorem”, Thesis University of Copenhagen, June 2007
20. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf
21. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf
22. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf
23. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010), http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf
24. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010), http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf
25. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010), http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf
26. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010), http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf
27. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010), http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf
28. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010), http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf
29. http://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf
30. http://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf
31. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010), http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf
32. D. Miličić, “Notes on Riemann Zeta Function”, http://www.math.utah.edu/~milicic/zeta.pdf P. Garrett, “Riemann’ Explicit/Exact formula”, (October 2, 2010), http://www.math.umn.edu/~garrett/m/mfms/notes_c/mfms_notes_02.pdf
33. http://math.postech.ac.kr/~sungpyo/Calculus1/Appendix.pdf
34. http://math.postech.ac.kr/~sungpyo/Calculus1/Appendix.pdf
35. http://math.postech.ac.kr/~sungpyo/Calculus1/Appendix.pdf
36. http://math.postech.ac.kr/~sungpyo/Calculus1/Appendix.pdf
37. http://math.postech.ac.kr/~sungpyo/Calculus1/Appendix.pdf