Lin Jiu

(RISC, Linz)

https://s3-seminar.github.io/seminars/lin-jiu

Title — The method of brackets (MOB) and integrating by differentiating (IBD)

Abstract — We introduce two methods of symbolic integration for definite integrals: the method of brackets, based on the Ramanujan’s master theorem; and integrating by differentiating method, from the Fourier transform of Dirac delta function. Besides some basic examples and latest results of both methods, respectively, a formal connection between these methods will be presented in the end.

Transcript

1. The method of brackets (MoB) Integration by Differentiating The Method of Brackets (MoB) and Integrating by Differentiating (IbD) Method Lin Jiu Research Institute for Symbolic Computation Johannes Kepler University Dec. 9th 2016 Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
2. The method of brackets (MoB) Integration by Differentiating Acknowledgement Joint Work with: V. H. Moll K. Kohl I. Gonzalez C. Vignat Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
3. The method of brackets (MoB) Integration by Differentiating Outlines 1 The method of brackets (MoB) Rules Ramanujan’s Master Theorem (RMT) Examples Recent result 2 Integration by Differentiating Formulas Recent proofs Connection Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
4. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Idea MoB evaluates the definite integral ∞ 0 f (x) dx (most of the time) in terms of SERIES, with ONLY SIX rules: Defintion [Indicator] φn := (−1)n n! = (−1)n Γ (n + 1) and φ1,...,r := φn1,...,nr = φn1 φn2 · · · φnr = r i=1 φni . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
5. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Idea MoB evaluates the definite integral ∞ 0 f (x) dx (most of the time) in terms of SERIES, with ONLY SIX rules: Defintion [Indicator] φn := (−1)n n! = (−1)n Γ (n + 1) and φ1,...,r := φn1,...,nr = φn1 φn2 · · · φnr = r i=1 φni . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
6. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Idea MoB evaluates the definite integral ∞ 0 f (x) dx (most of the time) in terms of SERIES, with ONLY SIX rules: Defintion [Indicator] φn := (−1)n n! = (−1)n Γ (n + 1) and φ1,...,r := φn1,...,nr = φn1 φn2 · · · φnr = r i=1 φni . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
7. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Idea MoB evaluates the definite integral ∞ 0 f (x) dx (most of the time) in terms of SERIES, with ONLY SIX rules: Defintion [Indicator] φn := (−1)n n! = (−1)n Γ (n + 1) and φ1,...,r := φn1,...,nr = φn1 φn2 · · · φnr = r i=1 φni . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
8. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Idea MoB evaluates the definite integral ∞ 0 f (x) dx (most of the time) in terms of SERIES, with ONLY SIX rules: Defintion [Indicator] φn := (−1)n n! = (−1)n Γ (n + 1) and φ1,...,r := φn1,...,nr = φn1 φn2 · · · φnr = r i=1 φni . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
9. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules (P-Production; E-Evaluation) I = ∞ 0 f (x) dx P1: f (x) = ∞ n=0 anxαn+β−1 ⇒ ∞ 0 f (x) dx → n an αn + β Bracket Series; P2: For α < 0, (a1 + · · · + ar )α → n1,...,nr φ1,...,r an1 1 · · · anr r −α+n1+···+nr Γ(−α) ; P3: For each bracket series, we assign index=# of sums− # of brackets; E1: n φnf (n) αn + β = 1 |α| f (n∗) Γ (−n∗), where n∗ solves αn + β = 0; E2: n1,...,nr φ1,...,r f (n1, . . . , nr ) r i=1 ai1n1 + · · · + air nr + ci = f (n∗ 1 ,...,n∗ r ) r i=1 Γ(−n∗ i ) |det A| , (n∗ 1 , . . . , n∗ r ) solves    a11n1 + · · · + a1r nr + c1 = 0 . . . · · · ar1n1 + · · · + arr nr + cr = 0 ; E3: The value of a multi-dimensional bracket series of POSITIVE index is obtained by computing all the contributions of maximal rank by Rule E2. These contributions to the integral appear as series in the free parameters. Series converging in a common region are added and divergent series are discarded. Any series producing a non-real contribution is also discarded. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
10. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Ramanujan’s Master Theorem[RMT] Theorem[RMT] ∞ 0 xs−1 a (0) − a (1) 1! x + a (2) 2! x2 − · · · dx = a (−s) Γ (s) (1) ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (2) [Hardy] •H (δ) := {s = σ + ιt : σ ≥ −δ, 0 < δ < 1}; •ψ (x) ∈ C∞ (H (δ)); ∃C, P, A, A < π such that |ψ (s)| ≤ CePδ+A|t|, ∀s ∈ H (δ); •0 < c < δ, Ψ (x) := 1 2πι c+ι∞ c−ι∞ π sin(πs) ψ (−s) x−s dx 0<x<e−P ==== ∞ k=0 ψ (k) (−x)k ; ∞ 0 Ψ (x) xs−1dx = π sin (πs) ψ (−s) . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
11. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Ramanujan’s Master Theorem[RMT] Theorem[RMT] ∞ 0 xs−1 a (0) − a (1) 1! x + a (2) 2! x2 − · · · dx = a (−s) Γ (s) (1) ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (2) [Hardy] •H (δ) := {s = σ + ιt : σ ≥ −δ, 0 < δ < 1}; •ψ (x) ∈ C∞ (H (δ)); ∃C, P, A, A < π such that |ψ (s)| ≤ CePδ+A|t|, ∀s ∈ H (δ); •0 < c < δ, Ψ (x) := 1 2πι c+ι∞ c−ι∞ π sin(πs) ψ (−s) x−s dx 0<x<e−P ==== ∞ k=0 ψ (k) (−x)k ; ∞ 0 Ψ (x) xs−1dx = π sin (πs) ψ (−s) . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
12. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Ramanujan’s Master Theorem[RMT] Theorem[RMT] ∞ 0 xs−1 a (0) − a (1) 1! x + a (2) 2! x2 − · · · dx = a (−s) Γ (s) (1) ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (2) [Hardy] •H (δ) := {s = σ + ιt : σ ≥ −δ, 0 < δ < 1}; •ψ (x) ∈ C∞ (H (δ)); ∃C, P, A, A < π such that |ψ (s)| ≤ CePδ+A|t|, ∀s ∈ H (δ); •0 < c < δ, Ψ (x) := 1 2πι c+ι∞ c−ι∞ π sin(πs) ψ (−s) x−s dx 0<x<e−P ==== ∞ k=0 ψ (k) (−x)k ; ∞ 0 Ψ (x) xs−1dx = π sin (πs) ψ (−s) . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
13. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Ramanujan’s Master Theorem[RMT] Theorem[RMT] ∞ 0 xs−1 a (0) − a (1) 1! x + a (2) 2! x2 − · · · dx = a (−s) Γ (s) (1) ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (2) [Hardy] •H (δ) := {s = σ + ιt : σ ≥ −δ, 0 < δ < 1}; •ψ (x) ∈ C∞ (H (δ)); ∃C, P, A, A < π such that |ψ (s)| ≤ CePδ+A|t|, ∀s ∈ H (δ); •0 < c < δ, Ψ (x) := 1 2πι c+ι∞ c−ι∞ π sin(πs) ψ (−s) x−s dx 0<x<e−P ==== ∞ k=0 ψ (k) (−x)k ; ∞ 0 Ψ (x) xs−1dx = π sin (πs) ψ (−s) . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
14. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Ramanujan’s Master Theorem[RMT] Theorem[RMT] ∞ 0 xs−1 a (0) − a (1) 1! x + a (2) 2! x2 − · · · dx = a (−s) Γ (s) (1) ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (2) [Hardy] •H (δ) := {s = σ + ιt : σ ≥ −δ, 0 < δ < 1}; •ψ (x) ∈ C∞ (H (δ)); ∃C, P, A, A < π such that |ψ (s)| ≤ CePδ+A|t|, ∀s ∈ H (δ); •0 < c < δ, Ψ (x) := 1 2πι c+ι∞ c−ι∞ π sin(πs) ψ (−s) x−s dx 0<x<e−P ==== ∞ k=0 ψ (k) (−x)k ; ∞ 0 Ψ (x) xs−1dx = π sin (πs) ψ (−s) . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
15. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again Theorem[RMT] ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (1) Integrand→Power Series; (2) Keep Track of s; (3) Apply the Formula; (4) Multiple Integrals; ∞ 0 ∞ 0 n,m a (m, n) xmyndxdy =? (5) More Sums than Integrals (brackets); ∞ 0 f1 (x) f2 (x) dx = ∞ 0 m,n a (m, n) xm+ndx = m,n a (m, n) m + n + 1 =? (6) Extra. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
16. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again Theorem[RMT] ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (1) Integrand→Power Series; (2) Keep Track of s; (3) Apply the Formula; (4) Multiple Integrals; ∞ 0 ∞ 0 n,m a (m, n) xmyndxdy =? (5) More Sums than Integrals (brackets); ∞ 0 f1 (x) f2 (x) dx = ∞ 0 m,n a (m, n) xm+ndx = m,n a (m, n) m + n + 1 =? (6) Extra. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
17. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again Theorem[RMT] ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (1) Integrand→Power Series; (2) Keep Track of s; (3) Apply the Formula; (4) Multiple Integrals; ∞ 0 ∞ 0 n,m a (m, n) xmyndxdy =? (5) More Sums than Integrals (brackets); ∞ 0 f1 (x) f2 (x) dx = ∞ 0 m,n a (m, n) xm+ndx = m,n a (m, n) m + n + 1 =? (6) Extra. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
18. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again P1 : f (x) = ∞ n=0 an xαn+β−1 ⇒ ∞ 0 f (x) dx → n an αn + β s − 1 → s P2 : For α < 0,(a1 + · · · + ar )α → n1,...,nr φ1,...,r an1 1 · · · anr r −α+n1+···+nr Γ(−α) ; P3 : Index=# of sums− # of brackets; Just a definition E1 : n φn f (n) αn + β = f (n∗)Γ(−n∗) |α| ,, n∗ solves αn + β = 0; RMT E2 : Iteration of RMT n1,...,nr φ1,...,r f (n1, . . . , nr ) r i=1 ai1 n1 + · · · + air nr + ci = f (n∗ 1 , . . . , n∗ r ) r i=1 Γ (−n∗ i ) |det A| E3 : The value of a multi-dimensional bracket series of POSITIVE index is obtained by computing all the contributions of maximal rank by Rule E2. These contributions to the integral appear as series in the free parameters. Series converging in a common region are added and divergent series are discarded. Any series producing a non-real contribution is also discarded. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
19. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again Theorem[RMT] ∞ 0 xs−1 ∞ n=0 φn a (n) xn dx = a (−s) Γ (s) (1) Integrand→Power Series; (2) Keep Track of s; (3) Apply the Formula; (4) Multiple Integrals; ∞ 0 ∞ 0 n,m a (m, n) xmyndxdy =? (5) More Sums than Integrals (brackets); ∞ 0 f1 (x) f2 (x) dx = ∞ 0 m,n a (m, n) xm+ndx = m,n a (m, n) m + n + 1 =? (6) Extra. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
20. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again P1 : f (x) = ∞ n=0 an xαn+β−1 ⇒ ∞ 0 f (x) dx → n an αn + β s − 1 → s P2 :For α < 0, (a1 + · · · + ar )α → n1,...,nr φ1,...,r an1 1 · · · anr r −α+n1+···+nr Γ(−α) ; P3 : Index=# of sums− # of brackets; Just a definition E1 : n φn f (n) αn + β = f (n∗)Γ(−n∗) |α| , n∗ solves αn + β = 0; RMT E2 : Iteration of RMT n1,...,nr φ1,...,r f (n1, . . . , nr ) r i=1 ai1 n1 + · · · + air nr + ci = f (n∗ 1 , . . . , n∗ r ) r i=1 Γ (−n∗ i ) |det A| E3 : The value of a multi-dimensional bracket series of POSITIVE index is obtained by computing all the contributions of maximal rank by Rule E2. These contributions to the integral appear as series in the free parameters. Series converging in a common region are added and divergent series are discarded. Any series producing a non-real contribution is also discarded. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
21. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again Theorem[RMT] ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (1) Integrand→Power Series; (2) Keep Track of s; (3) Apply the Formula; (4) Multiple Integrals; ∞ 0 ∞ 0 n,m a (m, n) xmyndxdy =? (5) More Sums than Integrals (brackets); ∞ 0 f1 (x) f2 (x) dx = ∞ 0 m,n a (m, n) xm+ndx = m,n a (m, n) m + n + 1 =? (6) Extra. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
22. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again P1 : f (x) = ∞ n=0 an xαn+β−1 ⇒ ∞ 0 f (x) dx → n an αn + β s − 1 → s P2 :For α < 0, (a1 + · · · + ar )α → n1,...,nr φ1,...,r an1 1 · · · anr r −α+n1+···+nr Γ(−α) ; P3 : Index=# of sums− # of brackets; Just a definition E1 : n φn f (n) αn + β = f (n∗)Γ(−n∗) |α| , n∗ solves αn + β = 0; RMT E2 : Iteration of RMT n1,...,nr φ1,...,r f (n1, . . . , nr ) r i=1 ai1 n1 + · · · + air nr + ci = f (n∗ 1 , . . . , n∗ r ) r i=1 Γ (−n∗ i ) |det A| E3 : The value of a multi-dimensional bracket series of POSITIVE index is obtained by computing all the contributions of maximal rank by Rule E2. These contributions to the integral appear as series in the free parameters. Series converging in a common region are added and divergent series are discarded. Any series producing a non-real contribution is also discarded. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
23. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again Theorem[RMT] ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (1) Integrand→Power Series; (2) Keep Track of s; (3) Apply the Formula; (4) Multiple Integrals; ∞ 0 ∞ 0 n,m a (m, n) xmyndxdy =? (5) More Sums than Integrals (brackets); ∞ 0 f1 (x) f2 (x) dx = ∞ 0 m,n a (m, n) xm+ndx = m,n a (m, n) m + n + 1 =? (6) Extra. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
24. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again P1 : f (x) = ∞ n=0 an xαn+β−1 ⇒ ∞ 0 f (x) dx → n an αn + β s − 1 → s P2 :For α < 0, (a1 + · · · + ar )α → n1,...,nr φ1,...,r an1 1 · · · anr r −α+n1+···+nr Γ(−α) ; P3 : Index=# of sums− # of brackets; Just a definition E1 : n φn f (n) αn + β = f (n∗)Γ(−n∗) |α| , n∗ solves αn + β = 0; RMT E2 : Iteration of RMT n1,...,nr φ1,...,r f (n1, . . . , nr ) r i=1 ai1 n1 + · · · + air nr + ci = f (n∗ 1 , . . . , n∗ r ) r i=1 Γ (−n∗ i ) |det A| E3 : The value of a multi-dimensional bracket series of POSITIVE index is obtained by computing all the contributions of maximal rank by Rule E2. These contributions to the integral appear as series in the free parameters. Series converging in a common region are added and divergent series are discarded. Any series producing a non-real contribution is also discarded. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
25. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again P1 : f (x) = ∞ n=0 an xαn+β−1 ⇒ ∞ 0 f (x) dx → n an αn + β s − 1 → s P2 :For α < 0, (a1 + · · · + ar )α → n1,...,nr φ1,...,r an1 1 · · · anr r −α+n1+···+nr Γ(−α) ; P3 : Index=# of sums− # of brackets; Just a definition E1 : n φn f (n) αn + β = f (n∗)Γ(−n∗) |α| , n∗ solves αn + β = 0; RMT E2 : Iteration of RMT n1,...,nr φ1,...,r f (n1, . . . , nr ) r i=1 ai1 n1 + · · · + air nr + ci = f (n∗ 1 , . . . , n∗ r ) r i=1 Γ (−n∗ i ) |det A| E3 : The value of a multi-dimensional bracket series of POSITIVE index is obtained by computing all the contributions of maximal rank by Rule E2. These contributions to the integral appear as series in the free parameters. Series converging in a common region are added and divergent series are discarded. Any series producing a non-real contribution is also discarded. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
26. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again Theorem[RMT] ∞ 0 xs−1 ∞ n=0 φna (n) xn dx = a (−s) Γ (s) (1) Integrand→Power Series; (2) Keep Track of s; (3) Apply the Formula; (4) Multiple Integrals; ∞ 0 ∞ 0 n,m a (m, n) xmyndxdy =? (5) More Sums than Integrals (brackets); ∞ 0 f1 (x) f2 (x) dx = ∞ 0 m,n a (m, n) xm+ndx = m,n a (m, n) m + n + 1 =? (6) Extra. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
27. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rules Again P1 : f (x) = ∞ n=0 an xαn+β−1 ⇒ ∞ 0 f (x) dx → n an αn + β s − 1 → s P2 :For α < 0, (a1 + · · · + ar )α → n1,...,nr φ1,...,r an1 1 · · · anr r −α+n1+···+nr Γ(−α) ; P3 : Index=# of sums− # of brackets; Just a definition E1 : n φn f (n) αn + β = f (n∗)Γ(−n∗) |α| , n∗ solves αn + β = 0; RMT E2 : Iteration of RMT n1,...,nr φ1,...,r f (n1, . . . , nr ) r i=1 ai1 n1 + · · · + air nr + ci = f (n∗ 1 , . . . , n∗ r ) r i=1 Γ (−n∗ i ) |det A| E3 : The value of a multi-dimensional bracket series of POSITIVE index is obtained by computing all the contributions of maximal rank by Rule E2. These contributions to the integral appear as series in the free parameters. Series converging in a common region are added and divergent series are discarded. Any series producing a non-real contribution is also discarded. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
28. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Rule P2 Γ (−α) (a1 + · · · + ar )−α . = ∞ 0 x−α−1e−(a1+···+ar )x dx = ∞ 0 x−α−1e−a1x e−a2x · · · e−ar x dx = ∞ 0 x−α−1 r i=1   ∞ ni =0 φni (ax)ni   dx = ∞ 0 n1,...,nr φ1,...,r an1 1 · · · anr r xn1+···+nr −α−1dx = n1,...,nr φ1,...,r an1 1 · · · anr r −α + n1 + · · · + nr Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
29. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay [y > 0 Re(a) > 0] Rule P2 : 1 √ a2 + x2 = a2 + x2 − 1 2 = n1,n2 φ1,2 a2n1 x2n2 1 2 + n1 + n2 Γ 1 2 J0 (xy) J0 (xy) = n3 φn3 y2n3 Γ (n3 + 1) 22n3 x2n3 Rule P1 I = ∞ 0 n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 x2n2+2n3+1dx = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
30. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay [y > 0 Re(a) > 0] Rule P2 : 1 √ a2 + x2 = a2 + x2 − 1 2 = n1,n2 φ1,2 a2n1 x2n2 1 2 + n1 + n2 Γ 1 2 J0 (xy) J0 (xy) = n3 φn3 y2n3 Γ (n3 + 1) 22n3 x2n3 Rule P1 I = ∞ 0 n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 x2n2+2n3+1dx = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
31. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay [y > 0 Re(a) > 0] Rule P2 : 1 √ a2 + x2 = a2 + x2 − 1 2 = n1,n2 φ1,2 a2n1 x2n2 1 2 + n1 + n2 Γ 1 2 J0 (xy) J0 (xy) = n3 φn3 y2n3 Γ (n3 + 1) 22n3 x2n3 Rule P1 I = ∞ 0 n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 x2n2+2n3+1dx = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
32. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay [y > 0 Re(a) > 0] Rule P2 : 1 √ a2 + x2 = a2 + x2 − 1 2 = n1,n2 φ1,2 a2n1 x2n2 1 2 + n1 + n2 Γ 1 2 J0 (xy) J0 (xy) = n3 φn3 y2n3 Γ (n3 + 1) 22n3 x2n3 Rule P1 I = ∞ 0 n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 x2n2+2n3+1dx = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
33. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay [y > 0 Re(a) > 0] Rule P2 : 1 √ a2 + x2 = a2 + x2 − 1 2 = n1,n2 φ1,2 a2n1 x2n2 1 2 + n1 + n2 Γ 1 2 J0 (xy) J0 (xy) = n3 φn3 y2n3 Γ (n3 + 1) 22n3 x2n3 Rule P1 I = ∞ 0 n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 x2n2+2n3+1dx = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
34. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay I = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 ; n1 free: n∗ 2 = − 1 2 − n1; n∗ 3 = − 1 2 + n1; det = 2: I = 1 2 n1 φn1 y2n1−1a2n1 Γ n1 + 1 2 Γ 1 2 22n1−1 Γ n1 + 1 2 Γ −n1 + 1 2 = 1 y ∞ n1=0 φn1 ay 2 2n1 Γ 1 2 − n1 Γ 1 2 = 1 y cosh (ay) ; n2 free : I = 1 √ πy ∞ n2=0 Γ n2+ 1 2 Γ(−n2) 2 ay 2n2+1 = 0; n3 free : I = Series = − sinh(ay) y ; E3 : I = 1 y cosh (ay) − sinh (ay) y = y−1e−ay . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
35. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay I = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 ; n1 free: n∗ 2 = − 1 2 − n1; n∗ 3 = − 1 2 + n1; det = 2: I = 1 2 n1 φn1 y2n1−1a2n1 Γ n1 + 1 2 Γ 1 2 22n1−1 Γ n1 + 1 2 Γ −n1 + 1 2 = 1 y ∞ n1=0 φn1 ay 2 2n1 Γ 1 2 − n1 Γ 1 2 = 1 y cosh (ay) ; n2 free : I = 1 √ πy ∞ n2=0 Γ n2+ 1 2 Γ(−n2) 2 ay 2n2+1 = 0; n3 free : I = Series = − sinh(ay) y ; E3 : I = 1 y cosh (ay) − sinh (ay) y = y−1e−ay . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
36. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay I = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 ; n1 free: n∗ 2 = − 1 2 − n1; n∗ 3 = − 1 2 + n1; det = 2: I = 1 2 n1 φn1 y2n1−1a2n1 Γ n1 + 1 2 Γ 1 2 22n1−1 Γ n1 + 1 2 Γ −n1 + 1 2 = 1 y ∞ n1=0 φn1 ay 2 2n1 Γ 1 2 − n1 Γ 1 2 = 1 y cosh (ay) ; n2 free : I = 1 √ πy ∞ n2=0 Γ n2+ 1 2 Γ(−n2) 2 ay 2n2+1 = 0; n3 free : I = Series = − sinh(ay) y ; E3 : I = 1 y cosh (ay) − sinh (ay) y = y−1e−ay . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
37. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay I = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 ; n1 free: n∗ 2 = − 1 2 − n1; n∗ 3 = − 1 2 + n1; det = 2: I = 1 2 n1 φn1 y2n1−1a2n1 Γ n1 + 1 2 Γ 1 2 22n1−1 Γ n1 + 1 2 Γ −n1 + 1 2 = 1 y ∞ n1=0 φn1 ay 2 2n1 Γ 1 2 − n1 Γ 1 2 = 1 y cosh (ay) ; n2 free : I = 1 √ πy ∞ n2=0 Γ n2+ 1 2 Γ(−n2) 2 ay 2n2+1 = 0; n3 free : I = Series = − sinh(ay) y ; E3 : I = 1 y cosh (ay) − sinh (ay) y = y−1e−ay . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
38. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay I = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 ; n1 free: n∗ 2 = − 1 2 − n1; n∗ 3 = − 1 2 + n1; det = 2: I = 1 2 n1 φn1 y2n1−1a2n1 Γ n1 + 1 2 Γ 1 2 22n1−1 Γ n1 + 1 2 Γ −n1 + 1 2 = 1 y ∞ n1=0 φn1 ay 2 2n1 Γ 1 2 − n1 Γ 1 2 = 1 y cosh (ay) ; n2 free : I = 1 √ πy ∞ n2=0 Γ n2+ 1 2 Γ(−n2) 2 ay 2n2+1 = 0; n3 free : I = Series = − sinh(ay) y ; E3 : I = 1 y cosh (ay) − sinh (ay) y = y−1e−ay . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
39. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay I = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 ; n1 free: n∗ 2 = − 1 2 − n1; n∗ 3 = − 1 2 + n1; det = 2: I = 1 2 n1 φn1 y2n1−1a2n1 Γ n1 + 1 2 Γ 1 2 22n1−1 Γ n1 + 1 2 Γ −n1 + 1 2 = 1 y ∞ n1=0 φn1 ay 2 2n1 Γ 1 2 − n1 Γ 1 2 = 1 y cosh (ay) ; n2 free : I = 1 √ πy ∞ n2=0 Γ n2+ 1 2 Γ(−n2) 2 ay 2n2+1 = 0; n3 free : I = Series = − sinh(ay) y ; E3 : I = 1 y cosh (ay) − sinh (ay) y = y−1e−ay . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
40. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay I = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 ; n1 free: n∗ 2 = − 1 2 − n1; n∗ 3 = − 1 2 + n1; det = 2: I = 1 2 n1 φn1 y2n1−1a2n1 Γ n1 + 1 2 Γ 1 2 22n1−1 Γ n1 + 1 2 Γ −n1 + 1 2 = 1 y ∞ n1=0 φn1 ay 2 2n1 Γ 1 2 − n1 Γ 1 2 = 1 y cosh (ay) ; n2 free : I = 1 √ πy ∞ n2=0 Γ n2+ 1 2 Γ(−n2) 2 ay 2n2+1 = 0; n3 free : I = Series = − sinh(ay) y ; E3 : I = 1 y cosh (ay) − sinh (ay) y = y−1e−ay . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
41. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay I = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 ; n1 free: n∗ 2 = − 1 2 − n1; n∗ 3 = − 1 2 + n1; det = 2: I = 1 2 n1 φn1 y2n1−1a2n1 Γ n1 + 1 2 Γ 1 2 22n1−1 Γ n1 + 1 2 Γ −n1 + 1 2 = 1 y ∞ n1=0 φn1 ay 2 2n1 Γ 1 2 − n1 Γ 1 2 = 1 y cosh (ay) ; n2 free : I = 1 √ πy ∞ n2=0 Γ n2+ 1 2 Γ(−n2) 2 ay 2n2+1 = 0; n3 free : I = Series = − sinh(ay) y ; E3 : I = 1 y cosh (ay) − sinh (ay) y = y−1e−ay . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
42. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Examples I := ∞ 0 xJ0(xy) dx √ a2 + x2 = y−1e−ay I = n1,n2,n3 φ1,2,3 y2n3 a2n1 Γ (n3 + 1) Γ 1 2 22n3 n1 + n2 + 1 2 2n2 + 2n3 + 2 ; n1 free: n∗ 2 = − 1 2 − n1; n∗ 3 = − 1 2 + n1; det = 2: I = 1 2 n1 φn1 y2n1−1a2n1 Γ n1 + 1 2 Γ 1 2 22n1−1 Γ n1 + 1 2 Γ −n1 + 1 2 = 1 y ∞ n1=0 φn1 ay 2 2n1 Γ 1 2 − n1 Γ 1 2 = 1 y cosh (ay) ; n2 free : I = 1 √ πy ∞ n2=0 Γ n2+ 1 2 Γ(−n2) 2 ay 2n2+1 = 0; n3 free : I = Series = − sinh(ay) y ; E3 : I = 1 y cosh (ay) − sinh (ay) y = y−1e−ay . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
43. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim I = Rm f x2 1 + · · · + x2 m dx1 · · · dxm (I) Usual method: By the n-dim spherical coordinate that r = x2 1 + · · · + x2 m and                  x1 = r cos (φ1 ) , 0 ≤ φ1 ≤ π, x2 = r sin (φ2 ) cos (φ2 ) , 0 ≤ φ2 ≤ π, · · · · · · xn−2 = r sin (φ1 ) · · · sin (φm−3 ) cos (φm−2 ) , 0 ≤ φm−2 ≤ π, xn−1 = r sin (φ1 ) · · · sin (φm−2 ) cos (φm−1 ) , 0 ≤ φm−1 ≤ 2π, xn−1 = r sin (φ1 ) · · · sin (φm−2 ) sin (φm−1 ) , 0 ≤ r < ∞, we have dx1 · · · dxm = rm−1 sinm−2 (φ1 ) · · · sin (φm−2 ) drdφ1 · · · dφm−1. Thus, I = 2π m 2 ∞ 0 rm−1f r2 dr 1 Γ m 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
44. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim I = Rm f x2 1 + · · · + x2 m dx1 · · · dxm (I) Usual method: By the n-dim spherical coordinate that r = x2 1 + · · · + x2 m and                  x1 = r cos (φ1 ) , 0 ≤ φ1 ≤ π, x2 = r sin (φ2 ) cos (φ2 ) , 0 ≤ φ2 ≤ π, · · · · · · xn−2 = r sin (φ1 ) · · · sin (φm−3 ) cos (φm−2 ) , 0 ≤ φm−2 ≤ π, xn−1 = r sin (φ1 ) · · · sin (φm−2 ) cos (φm−1 ) , 0 ≤ φm−1 ≤ 2π, xn−1 = r sin (φ1 ) · · · sin (φm−2 ) sin (φm−1 ) , 0 ≤ r < ∞, we have dx1 · · · dxm = rm−1 sinm−2 (φ1 ) · · · sin (φm−2 ) drdφ1 · · · dφm−1. Thus, I = 2π m 2 ∞ 0 rm−1f r2 dr 1 Γ m 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
45. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim I = Rm f x2 1 + · · · + x2 m dx1 · · · dxm (I) Usual method: By the n-dim spherical coordinate that r = x2 1 + · · · + x2 m and                  x1 = r cos (φ1 ) , 0 ≤ φ1 ≤ π, x2 = r sin (φ2 ) cos (φ2 ) , 0 ≤ φ2 ≤ π, · · · · · · xn−2 = r sin (φ1 ) · · · sin (φm−3 ) cos (φm−2 ) , 0 ≤ φm−2 ≤ π, xn−1 = r sin (φ1 ) · · · sin (φm−2 ) cos (φm−1 ) , 0 ≤ φm−1 ≤ 2π, xn−1 = r sin (φ1 ) · · · sin (φm−2 ) sin (φm−1 ) , 0 ≤ r < ∞, we have dx1 · · · dxm = rm−1 sinm−2 (φ1 ) · · · sin (φm−2 ) drdφ1 · · · dφm−1. Thus, I = 2π m 2 ∞ 0 rm−1f r2 dr 1 Γ m 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
46. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim I = Rm f x2 1 + · · · + x2 m dx1 · · · dxm (I) Usual method: By the n-dim spherical coordinate that r = x2 1 + · · · + x2 m and                  x1 = r cos (φ1 ) , 0 ≤ φ1 ≤ π, x2 = r sin (φ2 ) cos (φ2 ) , 0 ≤ φ2 ≤ π, · · · · · · xn−2 = r sin (φ1 ) · · · sin (φm−3 ) cos (φm−2 ) , 0 ≤ φm−2 ≤ π, xn−1 = r sin (φ1 ) · · · sin (φm−2 ) cos (φm−1 ) , 0 ≤ φm−1 ≤ 2π, xn−1 = r sin (φ1 ) · · · sin (φm−2 ) sin (φm−1 ) , 0 ≤ r < ∞, we have dx1 · · · dxm = rm−1 sinm−2 (φ1 ) · · · sin (φm−2 ) drdφ1 · · · dφm−1. Thus, I = 2π m 2 ∞ 0 rm−1f r2 dr 1 Γ m 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
47. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim I = Rm f x2 1 + · · · + x2 m dx1 · · · dxm (II) The method of brackets: Suppose f (t) = ∞ l=0 φl a (l) tl , then, ∞ 0 rm−1f r2 dr = l φl a (l) 2l + m = l φl a (l) 2l + m = 1 2 a − m 2 Γ m 2 . So it suffices to show that I = 2π m 2 1 2 a − m 2 Γ −m 2 + 1 (−1)− m 2 Γ m 2 1 Γ m 2 = π m 2 a − m 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
48. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim I = Rm f x2 1 + · · · + x2 m dx1 · · · dxm (II) The method of brackets: Suppose f (t) = ∞ l=0 φl a (l) tl , then, ∞ 0 rm−1f r2 dr = l φl a (l) 2l + m = l φl a (l) 2l + m = 1 2 a − m 2 Γ m 2 . So it suffices to show that I = 2π m 2 1 2 a − m 2 Γ −m 2 + 1 (−1)− m 2 Γ m 2 1 Γ m 2 = π m 2 a − m 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
49. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim Direct computation shows: I = 2m Rm + ∞ l=0 φl a (l) x2 1 + · · · + x2 m l dx1 · · · dxm = 2m Rm + ∞ l=0 φl a (l) n1,...,nm n1+···+nm=l l n1, · · · , nm x2n1 1 · · · x2nm m dx1 · · · dxm = 2m l=n1+···+nm φl a (l) n1,...,nm φ1,...,m l n1, · · · , nm 1 φ1,...,m m j=1 2nj + 1 = AC... = π m 2 a − m 2 , as desired. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
50. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim Direct computation shows: I = 2m Rm + ∞ l=0 φl a (l) x2 1 + · · · + x2 m l dx1 · · · dxm = 2m Rm + ∞ l=0 φl a (l) n1,...,nm n1+···+nm=l l n1, · · · , nm x2n1 1 · · · x2nm m dx1 · · · dxm = 2m l=n1+···+nm φl a (l) n1,...,nm φ1,...,m l n1, · · · , nm 1 φ1,...,m m j=1 2nj + 1 = AC... = π m 2 a − m 2 , as desired. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
51. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim Direct computation shows: I = 2m Rm + ∞ l=0 φl a (l) x2 1 + · · · + x2 m l dx1 · · · dxm = 2m Rm + ∞ l=0 φl a (l) n1,...,nm n1+···+nm=l l n1, · · · , nm x2n1 1 · · · x2nm m dx1 · · · dxm = 2m l=n1+···+nm φl a (l) n1,...,nm φ1,...,m l n1, · · · , nm 1 φ1,...,m m j=1 2nj + 1 = AC... = π m 2 a − m 2 , as desired. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
52. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim I = Rm + xp1−1 1 · · · xpm−1 m dx1 · · · dxm (r0 + r1 x1 + · · · + rm xm )s = Γ (p1 ) · · · Γ (pm ) Γ (s − p1 − p2 − · · · − pn ) rp1 1 · · · rpm m rs−p1−···−pm 0 Γ (s) . (r0 + r1x1 + · · · + rmxm)−s = n0,n1,...,nm φ0,1,...,mrn0 0 rn1 1 xn1 1 · · · rnm m xnm m s + n0 + · · · + nm Γ (s) I = 1 Γ (s) n0,n1,...,nm φ0,1,...,mrn0 0 · · · rnm m s + n0 + · · · + nm m j=1 nm + pm .   1 1 1 · · · 1 0 1 0 · · · 0 · · · · · · · · · · · · · · · 0 0 0 0 1     n0 n1 · · · nm   +   s p1 · · · pm   =     0 0 0 · · · 0     , det A = 1, n∗ j = −pj , ∀j = 1, . . . , m and n∗ 0 = p1 + · · · + pm − s. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
53. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Multi-dim I = Rm + xp1−1 1 · · · xpm−1 m dx1 · · · dxm (r0 + r1 x1 + · · · + rm xm )s = Γ (p1 ) · · · Γ (pm ) Γ (s − p1 − p2 − · · · − pn ) rp1 1 · · · rpm m rs−p1−···−pm 0 Γ (s) . (r0 + r1x1 + · · · + rmxm)−s = n0,n1,...,nm φ0,1,...,mrn0 0 rn1 1 xn1 1 · · · rnm m xnm m s + n0 + · · · + nm Γ (s) I = 1 Γ (s) n0,n1,...,nm φ0,1,...,mrn0 0 · · · rnm m s + n0 + · · · + nm m j=1 nm + pm .   1 1 1 · · · 1 0 1 0 · · · 0 · · · · · · · · · · · · · · · 0 0 0 0 1     n0 n1 · · · nm   +   s p1 · · · pm   =     0 0 0 · · · 0     , det A = 1, n∗ j = −pj , ∀j = 1, . . . , m and n∗ 0 = p1 + · · · + pm − s. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
54. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Null/Divergent Series K0 (x) = ∞ 0 cos(tx)dt √ 1+t2 . K0 (x) = 1 2 n φnΓ (−n) x2n 4n and K0 (x) = n φn Γ n + 1 2 2 Γ (−n) · 4n x2n+1 M (K0) (s) = ∞ 0 xs−1K0 (x) dx = 2s−2Γ s 2 2 ∞ 0 xs−1K0 (x) dx = ∞ 0 xs−1 n φn Γ n + 1 2 2 Γ (−n) · 4n x2n+1 dx = n φn Γ n + 1 2 2 4n Γ (−n) s − 2n − 1 = 2s−2Γ s 2 2 , ∞ 0 xs−1K0 (x) dx = ∞ 0 xs−1 2 n φnΓ (−n) x2n 4n dx = 1 2 n φn Γ (−n) 4n 2n + s = 2s−2Γ s 2 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
55. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Null/Divergent Series K0 (x) = ∞ 0 cos(tx)dt √ 1+t2 . K0 (x) = 1 2 n φnΓ (−n) x2n 4n and K0 (x) = n φn Γ n + 1 2 2 Γ (−n) · 4n x2n+1 M (K0) (s) = ∞ 0 xs−1K0 (x) dx = 2s−2Γ s 2 2 ∞ 0 xs−1K0 (x) dx = ∞ 0 xs−1 n φn Γ n + 1 2 2 Γ (−n) · 4n x2n+1 dx = n φn Γ n + 1 2 2 4n Γ (−n) s − 2n − 1 = 2s−2Γ s 2 2 , ∞ 0 xs−1K0 (x) dx = ∞ 0 xs−1 2 n φnΓ (−n) x2n 4n dx = 1 2 n φn Γ (−n) 4n 2n + s = 2s−2Γ s 2 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
56. The method of brackets (MoB) Integration by Differentiating Rules Ramanujan’s Master Theorem (RMT) Examples Recent result Null/Divergent Series K0 (x) = ∞ 0 cos(tx)dt √ 1+t2 . K0 (x) = 1 2 n φnΓ (−n) x2n 4n and K0 (x) = n φn Γ n + 1 2 2 Γ (−n) · 4n x2n+1 M (K0) (s) = ∞ 0 xs−1K0 (x) dx = 2s−2Γ s 2 2 ∞ 0 xs−1K0 (x) dx = ∞ 0 xs−1 n φn Γ n + 1 2 2 Γ (−n) · 4n x2n+1 dx = n φn Γ n + 1 2 2 4n Γ (−n) s − 2n − 1 = 2s−2Γ s 2 2 , ∞ 0 xs−1K0 (x) dx = ∞ 0 xs−1 2 n φnΓ (−n) x2n 4n dx = 1 2 n φn Γ (−n) 4n 2n + s = 2s−2Γ s 2 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
57. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection DEF A. Kempf et al. study Dirac-delta function to obtain the following formulas: b a f (x) dx = lim ε→0 f (∂ε) eεb − eεa ε , ∞ −∞ f (x) dx = lim ε→0 2πf (−ι∂ε) δ (ε) = 2πδ (ι∂ε) f (ε) , ∞ 0 f (x) dx = lim ε→0 f (−∂ε) 1 ε , 0 −∞ f (x) dx = lim ε→0 f (∂ε) 1 ε , ∞ −∞ f (x) dx = lim ε→0 [f (−∂ε) + f (∂ε)] 1 ε , where ∂ε denotes the derivative with respect to ε. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
58. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection DEF A. Kempf et al. study Dirac-delta function to obtain the following formulas: b a f (x) dx = lim ε→0 f (∂ε) eεb − eεa ε , ∞ −∞ f (x) dx = lim ε→0 2πf (−ι∂ε) δ (ε) = 2πδ (ι∂ε) f (ε) , ∞ 0 f (x) dx = lim ε→0 f (−∂ε) 1 ε , 0 −∞ f (x) dx = lim ε→0 f (∂ε) 1 ε , ∞ −∞ f (x) dx = lim ε→0 [f (−∂ε) + f (∂ε)] 1 ε , where ∂ε denotes the derivative with respect to ε. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
59. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Example I = ∞ 0 sin x x dx = π 2 f (x) = 1 x · 1 2ι eιx − e−ιx I = lim ε→0 f (−∂ε ) 1 ε = 1 2ι lim ε→0 e−ι∂ε − eι∂ε 1 ∂ε ◦ 1 . Note that 1/∂ε is the inverse operation of derivative, i.e., integration. I = 1 2ι lim ε→0 e−ι∂ε − eι∂ε ◦ (ln ε + c) Recall that for the derivative operator ∂ε , so that ea∂ε ◦ f (ε) = f (ε + a) . I = 1 2ι lim ε→0 [(ln (ε − ι) + c) − (ln (ε + ι) + c)] = 1 2ι lim ε→0 [ln (ε − ι) − ln (ε + ι)] = 1 2ι −ιπ 2 − ιπ 2 = π 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
60. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Example I = ∞ 0 sin x x dx = π 2 f (x) = 1 x · 1 2ι eιx − e−ιx I = lim ε→0 f (−∂ε ) 1 ε = 1 2ι lim ε→0 e−ι∂ε − eι∂ε 1 ∂ε ◦ 1 . Note that 1/∂ε is the inverse operation of derivative, i.e., integration. I = 1 2ι lim ε→0 e−ι∂ε − eι∂ε ◦ (ln ε + c) Recall that for the derivative operator ∂ε , so that ea∂ε ◦ f (ε) = f (ε + a) . I = 1 2ι lim ε→0 [(ln (ε − ι) + c) − (ln (ε + ι) + c)] = 1 2ι lim ε→0 [ln (ε − ι) − ln (ε + ι)] = 1 2ι −ιπ 2 − ιπ 2 = π 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
61. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Example I = ∞ 0 sin x x dx = π 2 f (x) = 1 x · 1 2ι eιx − e−ιx I = lim ε→0 f (−∂ε ) 1 ε = 1 2ι lim ε→0 e−ι∂ε − eι∂ε 1 ∂ε ◦ 1 . Note that 1/∂ε is the inverse operation of derivative, i.e., integration. I = 1 2ι lim ε→0 e−ι∂ε − eι∂ε ◦ (ln ε + c) Recall that for the derivative operator ∂ε , so that ea∂ε ◦ f (ε) = f (ε + a) . I = 1 2ι lim ε→0 [(ln (ε − ι) + c) − (ln (ε + ι) + c)] = 1 2ι lim ε→0 [ln (ε − ι) − ln (ε + ι)] = 1 2ι −ιπ 2 − ιπ 2 = π 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
62. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Example I = ∞ 0 sin x x dx = π 2 f (x) = 1 x · 1 2ι eιx − e−ιx I = lim ε→0 f (−∂ε ) 1 ε = 1 2ι lim ε→0 e−ι∂ε − eι∂ε 1 ∂ε ◦ 1 . Note that 1/∂ε is the inverse operation of derivative, i.e., integration. I = 1 2ι lim ε→0 e−ι∂ε − eι∂ε ◦ (ln ε + c) Recall that for the derivative operator ∂ε , so that ea∂ε ◦ f (ε) = f (ε + a) . I = 1 2ι lim ε→0 [(ln (ε − ι) + c) − (ln (ε + ι) + c)] = 1 2ι lim ε→0 [ln (ε − ι) − ln (ε + ι)] = 1 2ι −ιπ 2 − ιπ 2 = π 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
63. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Example I = ∞ 0 sin x x dx = π 2 f (x) = 1 x · 1 2ι eιx − e−ιx I = lim ε→0 f (−∂ε ) 1 ε = 1 2ι lim ε→0 e−ι∂ε − eι∂ε 1 ∂ε ◦ 1 . Note that 1/∂ε is the inverse operation of derivative, i.e., integration. I = 1 2ι lim ε→0 e−ι∂ε − eι∂ε ◦ (ln ε + c) Recall that for the derivative operator ∂ε , so that ea∂ε ◦ f (ε) = f (ε + a) . I = 1 2ι lim ε→0 [(ln (ε − ι) + c) − (ln (ε + ι) + c)] = 1 2ι lim ε→0 [ln (ε − ι) − ln (ε + ι)] = 1 2ι −ιπ 2 − ιπ 2 = π 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
64. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Example I = ∞ 0 sin x x dx = π 2 f (x) = 1 x · 1 2ι eιx − e−ιx I = lim ε→0 f (−∂ε ) 1 ε = 1 2ι lim ε→0 e−ι∂ε − eι∂ε 1 ∂ε ◦ 1 . Note that 1/∂ε is the inverse operation of derivative, i.e., integration. I = 1 2ι lim ε→0 e−ι∂ε − eι∂ε ◦ (ln ε + c) Recall that for the derivative operator ∂ε , so that ea∂ε ◦ f (ε) = f (ε + a) . I = 1 2ι lim ε→0 [(ln (ε − ι) + c) − (ln (ε + ι) + c)] = 1 2ι lim ε→0 [ln (ε − ι) − ln (ε + ι)] = 1 2ι −ιπ 2 − ιπ 2 = π 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
65. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Example I = ∞ 0 sin x x dx = π 2 f (x) = 1 x · 1 2ι eιx − e−ιx I = lim ε→0 f (−∂ε ) 1 ε = 1 2ι lim ε→0 e−ι∂ε − eι∂ε 1 ∂ε ◦ 1 . Note that 1/∂ε is the inverse operation of derivative, i.e., integration. I = 1 2ι lim ε→0 e−ι∂ε − eι∂ε ◦ (ln ε + c) Recall that for the derivative operator ∂ε , so that ea∂ε ◦ f (ε) = f (ε + a) . I = 1 2ι lim ε→0 [(ln (ε − ι) + c) − (ln (ε + ι) + c)] = 1 2ι lim ε→0 [ln (ε − ι) − ln (ε + ι)] = 1 2ι −ιπ 2 − ιπ 2 = π 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
66. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Remark I = ∞ 0 1 x n φn Γ (n + 1) Γ (2n + 2) x2n+1dx = n φn Γ (n + 1) Γ (2n + 2) 2n + 1 = 1 2 · Γ −1 2 + 1 Γ 2 −1 2 + 2 Γ 1 2 = π 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
67. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Remark I = ∞ 0 1 x n φn Γ (n + 1) Γ (2n + 2) x2n+1dx = n φn Γ (n + 1) Γ (2n + 2) 2n + 1 = 1 2 · Γ −1 2 + 1 Γ 2 −1 2 + 2 Γ 1 2 = π 2 . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
68. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Proofs D. Jia , E. Tang, A. Kempf, “present a list of propositions that put the above integration by differentiation methods on a rigorous footing.” ∞ 0 f (x) e−xy dx = lim a→∞ f (−∂y ) 1 − e−ay y , provided that f : R → R is entire and Laplace transformable on R+. Formal/Key idea: ∞ 0 f (x) e−xy dx = ∞ 0 ∞ n=0 cnxne−xy dx = ∞ n=0 cn lim a→∞ a 0 xne−xy dx = ∞ n=0 cn lim a→∞ a 0 (−∂y )n e−xy dx. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
69. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Proofs D. Jia , E. Tang, A. Kempf, “present a list of propositions that put the above integration by differentiation methods on a rigorous footing.” ∞ 0 f (x) e−xy dx = lim a→∞ f (−∂y ) 1 − e−ay y , provided that f : R → R is entire and Laplace transformable on R+. Formal/Key idea: ∞ 0 f (x) e−xy dx = ∞ 0 ∞ n=0 cnxne−xy dx = ∞ n=0 cn lim a→∞ a 0 xne−xy dx = ∞ n=0 cn lim a→∞ a 0 (−∂y )n e−xy dx. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
70. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Proofs D. Jia , E. Tang, A. Kempf, “present a list of propositions that put the above integration by differentiation methods on a rigorous footing.” ∞ 0 f (x) e−xy dx = lim a→∞ f (−∂y ) 1 − e−ay y , provided that f : R → R is entire and Laplace transformable on R+. Formal/Key idea: ∞ 0 f (x) e−xy dx = ∞ 0 ∞ n=0 cnxne−xy dx = ∞ n=0 cn lim a→∞ a 0 xne−xy dx = ∞ n=0 cn lim a→∞ a 0 (−∂y )n e−xy dx. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
71. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Proofs D. Jia , E. Tang, A. Kempf, “present a list of propositions that put the above integration by differentiation methods on a rigorous footing.” ∞ 0 f (x) e−xy dx = lim a→∞ f (−∂y ) 1 − e−ay y , provided that f : R → R is entire and Laplace transformable on R+. Formal/Key idea: ∞ 0 f (x) e−xy dx = ∞ 0 ∞ n=0 cnxne−xy dx = ∞ n=0 cn lim a→∞ a 0 xne−xy dx = ∞ n=0 cn lim a→∞ a 0 (−∂y )n e−xy dx. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
72. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Proofs D. Jia , E. Tang, A. Kempf, “present a list of propositions that put the above integration by differentiation methods on a rigorous footing.” ∞ 0 f (x) e−xy dx = lim a→∞ f (−∂y ) 1 − e−ay y , provided that f : R → R is entire and Laplace transformable on R+. Formal/Key idea: ∞ 0 f (x) e−xy dx = ∞ 0 ∞ n=0 cnxne−xy dx = ∞ n=0 cn lim a→∞ a 0 xne−xy dx = ∞ n=0 cn lim a→∞ a 0 (−∂y )n e−xy dx. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
73. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Formal Connection f (x) = ∞ n=0 anxαn+β−1 ⇒ ∞ 0 f (x) dx = lim ε→0 ∞ 0 e−εx f (x) dx. lim ε→0 ∞ 0 e−εx f (x) dx = lim ε→0 ∞ 0 e−εx ∞ n=0 anxαn+β−1 dx = lim ε→0 ∞ n=0 an ∞ 0 e−εx xαn+β−1dx = lim ε→0 ∞ n=0 an ∞ 0 (−∂ε)αn+β−1 ◦ e−εx dx = lim ε→0 ∞ n=0 an (−∂ε)αn+β−1 ◦ ∞ 0 e−εx dx = lim ε→0 f (−∂ε) ◦ 1 ε . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
74. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Formal Connection f (x) = ∞ n=0 anxαn+β−1 ⇒ ∞ 0 f (x) dx = lim ε→0 ∞ 0 e−εx f (x) dx. lim ε→0 ∞ 0 e−εx f (x) dx = lim ε→0 ∞ 0 e−εx ∞ n=0 anxαn+β−1 dx = lim ε→0 ∞ n=0 an ∞ 0 e−εx xαn+β−1dx = lim ε→0 ∞ n=0 an ∞ 0 (−∂ε)αn+β−1 ◦ e−εx dx = lim ε→0 ∞ n=0 an (−∂ε)αn+β−1 ◦ ∞ 0 e−εx dx = lim ε→0 f (−∂ε) ◦ 1 ε . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
75. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Formal Connection f (x) = ∞ n=0 anxαn+β−1 ⇒ ∞ 0 f (x) dx = lim ε→0 ∞ 0 e−εx f (x) dx. lim ε→0 ∞ 0 e−εx f (x) dx = lim ε→0 ∞ 0 e−εx ∞ n=0 anxαn+β−1 dx = lim ε→0 ∞ n=0 an ∞ 0 e−εx xαn+β−1dx = lim ε→0 ∞ n=0 an ∞ 0 (−∂ε)αn+β−1 ◦ e−εx dx = lim ε→0 ∞ n=0 an (−∂ε)αn+β−1 ◦ ∞ 0 e−εx dx = lim ε→0 f (−∂ε) ◦ 1 ε . Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
76. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Formal Connection Recall P1 : f (x) = ∞ n=0 an xαn+β−1 ⇒ ∞ 0 f (x) dx = n an αn + β . Formally, a := ∞ 0 xa−1dx. and a ε := ∞ 0 xa−1e−εx dx ⇒ lim ε→0 a ε = a . Therefore n an αn + β = lim ε→0 n an αn + β ε = lim ε→0 n an ∞ 0 xαn+β−1e−εx dx = lim ε→0 ∞ 0 e−εx f (x) dx. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
77. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Formal Connection Recall P1 : f (x) = ∞ n=0 an xαn+β−1 ⇒ ∞ 0 f (x) dx = n an αn + β . Formally, a := ∞ 0 xa−1dx. and a ε := ∞ 0 xa−1e−εx dx ⇒ lim ε→0 a ε = a . Therefore n an αn + β = lim ε→0 n an αn + β ε = lim ε→0 n an ∞ 0 xαn+β−1e−εx dx = lim ε→0 ∞ 0 e−εx f (x) dx. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
78. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Formal Connection Recall P1 : f (x) = ∞ n=0 an xαn+β−1 ⇒ ∞ 0 f (x) dx = n an αn + β . Formally, a := ∞ 0 xa−1dx. and a ε := ∞ 0 xa−1e−εx dx ⇒ lim ε→0 a ε = a . Therefore n an αn + β = lim ε→0 n an αn + β ε = lim ε→0 n an ∞ 0 xαn+β−1e−εx dx = lim ε→0 ∞ 0 e−εx f (x) dx. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
79. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Formal Connection Recall P1 : f (x) = ∞ n=0 an xαn+β−1 ⇒ ∞ 0 f (x) dx = n an αn + β . Formally, a := ∞ 0 xa−1dx. and a ε := ∞ 0 xa−1e−εx dx ⇒ lim ε→0 a ε = a . Therefore n an αn + β = lim ε→0 n an αn + β ε = lim ε→0 n an ∞ 0 xαn+β−1e−εx dx = lim ε→0 ∞ 0 e−εx f (x) dx. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
80. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Possible Future Work Connect MoB to IbD, and provide rigorous proofs of the former. Ramanujan’s Master Theorem. Extension to other intervals. etc. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
81. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Possible Future Work Connect MoB to IbD, and provide rigorous proofs of the former. Ramanujan’s Master Theorem. Extension to other intervals. etc. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
82. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Possible Future Work Connect MoB to IbD, and provide rigorous proofs of the former. Ramanujan’s Master Theorem. Extension to other intervals. etc. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
83. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Possible Future Work Connect MoB to IbD, and provide rigorous proofs of the former. Ramanujan’s Master Theorem. Extension to other intervals. etc. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
84. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection Possible Future Work Connect MoB to IbD, and provide rigorous proofs of the former. Ramanujan’s Master Theorem. Extension to other intervals. etc. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
85. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection References B. C. Berndt, Ramanujan’s Notebooks Part I, Springer-Verlag, 1991. K. Boyadzhiev, V. H. Moll. The integrals in Gradshteyn and Ryzhik. Part 21: hyperbolic functions. Scientia. 22 (2013), 109–127, 2013. I. Gonzalez, K. Kohl, and V. H. Moll. Evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets. Scientia, 25 (2014), 65–84. I. Gonzalez and V. H. Moll. Definite integrals by the method of brackets. Part 1. Adv. Appl. Math., 45 (2010), 50–73. I. Gonzalez and I. Schmidt. Optimized negative dimensional integration method (NDIM) and multiloop Feynman diagram calculation. Nuclear Phys. B, 769 (2017), 124–173. I. Gonzalez and I. Schmidt. Modular application of an integration by fractional expansion (IBFE) method to multiloop Feynman diagrams. Phys. Rev. D, 78 (2008), 086003. I. G. Halliday and R. M. Ricotta. Negative dimensional integrals. I. Feynman graphs. Phys. Lett. B, 193 (1987), 241–246. I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products, Academic Press, 2015. G. H. Hardy. Ramanujan. Twelve Lectures on Subjects Suggested by His Life and Work. Chelsea Publishing Company, 1987. K. Kohl. Algorithmic Methods for Definite Integration. PhD thesis, Tulane University, 2011. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
86. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection References B. C. Berndt, Ramanujan’s Notebooks Part I, Springer-Verlag, 1991. K. Boyadzhiev, V. H. Moll. The integrals in Gradshteyn and Ryzhik. Part 21: hyperbolic functions. Scientia. 22 (2013), 109–127, 2013. I. Gonzalez, K. Kohl, and V. H. Moll. Evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets. Scientia, 25 (2014), 65–84. I. Gonzalez and V. H. Moll. Definite integrals by the method of brackets. Part 1. Adv. Appl. Math., 45 (2010), 50–73. I. Gonzalez and I. Schmidt. Optimized negative dimensional integration method (NDIM) and multiloop Feynman diagram calculation. Nuclear Phys. B, 769 (2017), 124–173. I. Gonzalez and I. Schmidt. Modular application of an integration by fractional expansion (IBFE) method to multiloop Feynman diagrams. Phys. Rev. D, 78 (2008), 086003. I. G. Halliday and R. M. Ricotta. Negative dimensional integrals. I. Feynman graphs. Phys. Lett. B, 193 (1987), 241–246. I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products, Academic Press, 2015. G. H. Hardy. Ramanujan. Twelve Lectures on Subjects Suggested by His Life and Work. Chelsea Publishing Company, 1987. K. Kohl. Algorithmic Methods for Definite Integration. PhD thesis, Tulane University, 2011. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
87. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection References A. Kempf, D. M. Jackson, and A. H. Morales, New Dirac Delta Function Based Methods with Applications to Perturbative Expansions in Quantum Field Theory, Journal of Physics A: Mathematical and Theoretical 47 (41): 415-204, 2014 A. Kempf, D. M. Jackson, and A. H. Morales, How to (Path-) Integrate by Differentiating, preprint, arXiv:math/1507.04348, 2015. D. Jia, E. Tang and A. Kempf, Integration by differentiation: new proofs, methods and examples, preprint, https://arxiv.org/abs/1610.09702, 2016. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
88. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection References A. Kempf, D. M. Jackson, and A. H. Morales, New Dirac Delta Function Based Methods with Applications to Perturbative Expansions in Quantum Field Theory, Journal of Physics A: Mathematical and Theoretical 47 (41): 415-204, 2014 A. Kempf, D. M. Jackson, and A. H. Morales, How to (Path-) Integrate by Differentiating, preprint, arXiv:math/1507.04348, 2015. D. Jia, E. Tang and A. Kempf, Integration by differentiation: new proofs, methods and examples, preprint, https://arxiv.org/abs/1610.09702, 2016. Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin
89. The method of brackets (MoB) Integration by Differentiating Formulas Recent proofs Connection End Thank you! Lin Jiu The Method of Brackets (MoB) and Integrating by Differentiatin