Matematikte, Jacobi dönüşümü bir integral dönüşümü matematikçinin adını almıştır Carl Gustav Jacob Jacobi, hangi kullanır Jacobi polinomları
dönüşümün çekirdekleri olarak.[1][2][3][4]
Bir fonksiyonun Jacobi dönüşümü
dır-dir[5]
![{ displaystyle J {F (x) } = f ^ { alpha, beta} (n) = int _ {- 1} ^ {1} (1-x) ^ { alpha} (1 + x) ^ { beta} P_ {n} ^ { alpha, beta} (x) F (x) dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b646649d2f81dab22534056c05c5067e3e24445)
Ters Jacobi dönüşümü şu şekilde verilir:
![{ displaystyle J ^ {- 1} {f ^ { alpha, beta} (n) } = F (x) = toplamı _ {n = 0} ^ { infty} { frac {1} { delta _ {n}}} f ^ { alpha, beta} (n) P_ {n} ^ { alpha, beta} (x), quad { text {nerede}} quad delta _ {n} = { frac {2 ^ { alpha + beta +1} Gama (n + alpha +1) Gama (n + beta +1)} {n! ( alpha + beta + 2n +1) Gama (n + alpha + beta +1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee28c500cc0469174c25f929c38e879de811853)
Bazı Jacobi dönüşüm çiftleri
![{ displaystyle F (x) ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef7c9dc516f87ed3e17287c60f65c0743b57a8f5) | ![{ displaystyle f ^ { alpha, beta} (n) ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/012177acd6877606a39aed1a3707609e34372cc4) |
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![{ displaystyle x ^ {m}, m <n ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/939564544d3ed363c80551a4d67da537edcf17c9) | ![{ displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{ displaystyle x ^ {n} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10f3d915f89551e1d02eb03af9b8b0a0a41622cc) | ![{ displaystyle n! ( alpha + beta + 2n + 1) delta _ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9fd22fa35f0c83a8f24e42515e7dce172809a7) |
![{ displaystyle P_ {m} ^ { alpha, beta} (x) ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc5bdf2aa597711448629e37ff68d7272e94ef9) | ![{ displaystyle delta _ {n} delta _ {mn}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4f11acbd43c150f2745d943a4f25b251b6ee2e) |
![{ displaystyle (1 + x) ^ {a- beta} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3b77564a7bcdff5e14de57982c59a4b4ad1dfa) | ![{ displaystyle { binom {n + alpha} {n}} 2 ^ { alpha + a + 1} { frac { Gamma (a + 1) Gamma ( alpha +1) Gamma (a- beta +1)} { Gama ( alpha + a + n + 2) Gama (a- beta + n + 1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c111a7a6e1dfb13401e762a3c40f83dc3c4cb5f4) |
![{ Displaystyle (1-x) ^ { sigma - alfa}, Re sigma> -1 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1af4ced7fbee441ee148fc80dbf797b76b0ddc5d) | ![{ displaystyle { frac {2 ^ { sigma + beta +1}} {n! Gama ( alfa - sigma)}} { frac { Gama ( sigma +1) Gama (n + beta +1) Gama ( alpha - sigma + n)} { Gama ( beta + sigma + n + 2)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbfd8ae9d39788bd0a48b3851b7714ba6e83e6c) |
![{ displaystyle (1-x) ^ { sigma - beta} P_ {m} ^ { alpha, sigma} (x), Re sigma> -1 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75ebd4cd9b313571d81f7d003d683dd0ba1d3cda) | ![{ displaystyle { frac {2 ^ { alpha + sigma +1}} {m! (nm)!}} { frac { Gama (n + alfa +1) Gama ( alfa + beta + m + n + 1) Gama ( sigma + m + 1) Gama ( alpha - beta +1)} { Gama ( alpha + beta + n + 1) Gama ( alpha + sigma + m + n + 2) Gama ( alpha - beta + m + 1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/969b4341e94414f6c20e2127c3c8c733909ff31e) |
![{ displaystyle 2 ^ { alpha + beta} Q ^ {- 1} (1-z + Q) ^ {- alpha} (1 + z + Q) ^ {- beta}, Q = (1 -2xz + z ^ {2}) ^ {1/2}, | z | <1 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0a8d6872939f7f4d6a49391193647dc7470b8fa) | ![{ displaystyle toplamı _ {n = 0} ^ { infty} delta _ {n} z ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69e432a30eea8f705c50683b07668e1d7340f548) |
![{ displaystyle (1-x) ^ {- alpha} (1 + x) ^ {- beta} { frac {d} {dx}} sol [(1-x) ^ { alpha +1} (1 + x) ^ { beta +1} { frac {d} {dx}} sağ] F (x) ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf3f023f7d443653e8a4664e7c2e71e76cb4e07) | ![{ displaystyle -n (n + alpha + beta +1) f ^ { alpha, beta} (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b39658e0796ed57c09d59fafed8ce116df62a25) |
![{ displaystyle sol {(1-x) ^ {- alpha} (1 + x) ^ {- beta} { frac {d} {dx}} sol [(1-x) ^ { alfa +1} (1 + x) ^ { beta +1} { frac {d} {dx}} right] right } ^ {k} F (x) ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f37708dcabf8079699227184c4cbfbcac88a868d) | ![{ displaystyle (-1) ^ {k} n ^ {k} (n + alpha + beta +1) ^ {k} f ^ { alpha, beta} (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0895414dbbc8d3f762bbebf2e9eeee336476f81a) |
Referanslar
- ^ Debnath, L. "Jacobi Dönüşümü Üzerine." Boğa. Cal. Matematik. Soc 55.3 (1963): 113-120.
- ^ Debnath, L. "KISMİ DİFERANSİYEL DENKLEMLERİN JACOBI DÖNÜŞÜMÜ İLE ÇÖZÜMÜ." CALCUTTA MATEMATİKSEL TOPLULUĞUN BÜLTENİ 59.3-4 (1967): 155.
- ^ Scott, E. J. "Jacobi dönüşümleri." (1953).
- ^ Shen, Jie; Wang, Yingwei; Xia, Jianlin (2019). "Hızlı yapılandırılmış Jacobi-Jacobi dönüşümleri". Matematik. Zorunlu. 88 (318): 1743–1772. doi:10.1090 / mcom / 3377.
- ^ Debnath, Lokenath ve Dambaru Bhatta. İntegral dönüşümler ve uygulamaları. CRC basın, 2014.