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Gamma Function and Beta Function Limits, Assignments of Mathematics

A.T. Still University of Health Sciences (ATSU)Mathematics

A series of mathematical problems and solutions related to the Gamma and Beta functions, including their limits and properties. It includes examples and formulas to calculate the Gamma and Beta functions for different values of n and m, and their relationship with other mathematical functions such as the integral and exponential functions.

Typology: Assignments

2020/2021

Uploaded on 03/29/2021

carlos-peguero 🇺🇸

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𝑬𝒋𝒆𝒎𝒑𝒍𝒐:𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑟:

1) Γ 7

2Γ 4 Γ 3 2) Γ 7/2

2Γ 3/2

Γ 𝑛 +1 = 𝑛!

Γ 7

2Γ 4 Γ 3 =Γ 6+1

2Γ 3+1 Γ 2+1 =6!

2 3! 2! =1.2.3.4.5.6

2 1.2.3 1.2 =30

Discover Assignments of Mathematics A.T. Still University of Health Sciences (ATSU)

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2 Γ 4 Γ 3 2)^

2 Γ 4 Γ 3 =^

2 3! 2! =^

2 Γ^

2 +^1

2 Γ^

2 +^1

2 Γ^

Γ 𝑛 = 𝑥𝑛−^1

∞ 0

𝑫𝒆𝒎𝒐𝒔𝒕𝒓𝒂𝒄𝒊ó𝒏:

Γ 1 / 2 = 𝑥^1 /^2 −^1

∞ 0

Γ 1 / 2 = 𝑥−^1 /^2

∞ 0

𝐻𝑎𝑐𝑖𝑒𝑛𝑑𝑜 𝑥 = 𝑢^2

Γ 1 / 2 = 𝑢−^1

∞ 0

𝑒−𝑢^2 2𝑢𝑑𝑢

𝐸𝑛 𝑙𝑜𝑠 𝑙𝑖𝑚𝑡𝑒𝑠 𝑠𝑒 𝑡𝑒𝑛𝑑𝑟á: 𝑥 = 0 𝑢 = 0

𝑥 = ∞ 𝑢 = ∞

Γ 1 / 2 = 2 𝑒−𝑢^2 𝑑𝑢

∞ 0

Análogamente:

Γ 1 / 2 = 2 𝑒−𝑣^2 𝑑𝑣

∞ 0

Γ 1 / 2 2 = 2 𝑒−𝑢^2 𝑑𝑢

∞ 0

2 𝑒−𝑣^2 𝑑𝑣

∞ 0 Γ 1 / 2 2 = 4 𝑒−^ 𝑢^2 +𝑣^2

∞ 0

𝐼𝑀^2 = lím 𝑀→∞ 𝑒−^ 𝑢^2 +𝑣^2

𝑀 0

𝐼^2 𝑀 = 𝑒

− 𝑢^2 +𝑣^2

𝑀 0

𝐼^2 𝑀 = 𝑒−^ 𝑢^2 +𝑣^2 𝑑𝑢𝑑𝑣

𝑅𝑀

𝑒−^ 𝑢^2 +𝑣^2 𝑑𝑢𝑑𝑣

𝑅 1

≤ 𝐼^2 𝑀 ≤ 𝑒−^ 𝑢^2 +𝑣^2 𝑑𝑢𝑑𝑣

𝑅 2

𝑆𝑖𝑒𝑛𝑑𝑜 𝑅 1 𝑦 𝑅 2 𝑙𝑎𝑠 𝑟𝑒𝑔𝑖𝑜𝑛𝑒𝑠 𝑑𝑒𝑙 𝑝𝑟𝑖𝑚𝑒𝑟 𝑐𝑢𝑎𝑑𝑟𝑎𝑛𝑡𝑒 𝑙𝑖𝑚𝑖𝑡𝑎𝑑𝑎𝑠 𝑝𝑜𝑟 𝑙𝑜𝑠 𝑐í𝑟𝑐𝑢𝑙𝑜𝑠 𝑑𝑒

𝑟𝑎𝑑𝑖𝑜 𝑀 𝑦 2𝑀 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑎𝑚𝑒𝑛𝑡𝑒. 𝑃𝑎𝑠𝑎𝑛𝑑𝑜 𝑎 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑝𝑜𝑙𝑎𝑟𝑒𝑠: 𝑢^2 + 𝑣^2 = 𝑟^2 0 ≤ 𝜙 ≤

𝑅 2 = 𝑏^2 + 𝑐^2 = 𝑀^2 + 𝑀^2 = 2𝑀^2

𝑅 2 = 2 𝑀^2 = 2 𝑀

Γ 1 / 2 2 = 𝐼𝑀^2 = 4 𝑒−^ 𝑢^2 +𝑣^2

𝑀 0

𝑥^2 𝑒−2𝑥^2 𝑑𝑥

∞ 0 𝐻𝑎𝑐𝑖𝑒𝑛𝑑𝑜 𝑢 = 2𝑥^2 𝑥^2 =

Γ 𝑛 = 𝑥𝑛−^1

∞ 0

𝐸𝑛 𝑙𝑜𝑠 𝑙𝑖𝑚𝑡𝑒𝑠 𝑠𝑒 𝑡𝑒𝑛𝑑𝑟á:

𝑥 = 0 𝑢 = 0

𝑥 = ∞ 𝑢 = ∞

𝐿𝑎 𝑓𝑢𝑛𝑐𝑖ó𝑛 𝐵𝑒𝑡𝑎 𝑑𝑒𝑛𝑜𝑡𝑎𝑑𝑎 𝑝𝑟 𝐵 𝑚, 𝑛 𝑠𝑒 𝑑𝑒𝑓𝑖𝑛𝑒 𝑝𝑜𝑟:

𝐵 𝑚, 𝑛 = 𝑥𝑚−1^ 1 − 𝑥 𝑛−1𝑑𝑥

1 0

𝐸𝑛 𝑙𝑜𝑠 𝑙𝑖𝑚𝑡𝑒𝑠 𝑠𝑒 𝑡𝑒𝑛𝑑𝑟á: 𝑥 = 0

𝐵 𝑚, 𝑛 = 𝑥𝑚−1^ 1 − 𝑥 𝑛−1𝑑𝑥

1 0

0 1

= 𝑦𝑛−1^ 1 − 𝑦 𝑚−1𝑑𝑦

1 0

𝐵 𝑚, 𝑛 = 𝑥𝑚−1^ 1 − 𝑥 𝑛−1𝑑𝑥

1 0

𝜋/ 0 𝑫𝒆𝒎𝒐𝒔𝒕𝒓𝒂𝒄𝒊ó𝒏:

𝐵 𝑚, 𝑛 = 𝑥𝑚−1^ 1 − 𝑥 𝑛−1𝑑𝑥 (1)

1 0 𝐻𝑎𝑐𝑖𝑒𝑛𝑑𝑜: 𝑥 = 𝑠𝑒𝑛^2 𝜃 𝐸𝑣𝑎𝑙𝑢𝑎𝑛𝑑𝑜 𝑙𝑜𝑠 𝑙𝑖𝑚𝑖𝑡𝑒𝑠: 𝑥 = 0

𝑥 = 1 𝜃 = 𝑎𝑟𝑐𝑠𝑒𝑛^ 1 =^ 𝑎𝑟𝑐𝑠𝑒𝑛1 =^ 𝜋/

𝐵 𝑚, 𝑛 = 𝑠𝑒𝑛^2 𝜃 𝑚−1^ 1 − 𝑠𝑒𝑛^2 𝜃 𝑛−

𝜋/ 0

𝑐𝑜𝑠^2 𝜃

𝜋/ 0

𝑫𝒆𝒎𝒐𝒔𝒕𝒓𝒂𝒄𝒊ó𝒏

𝑆𝑒𝑎 ∶ Γ 𝑚 = 𝑧𝑚−^1

∞ 0

𝐻𝑎𝑐𝑖𝑒𝑛𝑑𝑜: 𝑧 = 𝑥^2 𝑑𝑧 = 2𝑥𝑑𝑥

Γ 𝑚 = 𝑥^2 (𝑚−^1 )

∞ 0

𝑒−𝑥^2 2𝑥𝑑𝑥 = 2 𝑥2𝑚−^2 +^1

∞ 0

𝑒−𝑥^2 𝑑𝑥 = 2^ 𝑥2𝑚−^1

∞ 0

𝑒−𝑥^2 𝑑𝑥

𝐴𝑛á𝑙𝑜𝑔𝑎𝑚𝑒𝑛𝑡𝑒:

Γ 𝑚 = 2 𝑥2𝑚−^1

∞ 0

𝑒−𝑥^2 𝑑𝑥

Γ 𝑛 = 2 𝑦2𝑛−^1

∞ 0

𝑒−𝑦^2 𝑑𝑦

Γ 𝑚 Γ 𝑛 = 2 𝑥2𝑚−^1

∞ 0

𝑒−𝑥^2 𝑑𝑥 2 𝑦2𝑛−^1

∞ 0

𝑒−𝑦^2 𝑑𝑦

Γ 𝑚 Γ 𝑛 = 4 𝑥2𝑚−^1 𝑦^2 𝑛−^1 𝑒−^ 𝑥^2 +𝑦^2 𝑑𝑥𝑑𝑦

∞ 0

Γ 𝑚 Γ 𝑛 = 4 𝑥2𝑚−^1 𝑦^2 𝑛−^1 𝑒−^ 𝑥^2 +𝑦^2 𝑑𝑥𝑑𝑦

∞ 0

∞ 0 𝑃𝑎𝑠𝑎𝑛𝑑𝑜 𝑎 𝑐𝑜𝑜𝑟𝑑𝑒𝑛𝑎𝑑𝑎𝑠 𝑝𝑜𝑙𝑎𝑟𝑒𝑠:

𝑥 = 𝑟𝑐𝑜𝑠𝜙 𝑦 = 𝑟𝑠𝑒𝑛𝜙^ 𝑟^2 = 𝑥^2 + 𝑦^2

𝑅 𝑅

Γ 𝑚 Γ 𝑛 = 4 𝑟𝑐𝑜𝑠𝜙 2𝑚−^1 𝑟𝑠𝑒𝑛𝜙 2 𝑛−^1 𝑒−𝑟^2 𝑟𝑑𝑟𝑑𝑑𝜙

∞ 0

𝜋/ 2 0

Γ 𝑚 Γ 𝑛 = 4 𝑟𝑐𝑜𝑠𝜙 2𝑚−^1 𝑟𝑠𝑒𝑛𝜙 2 𝑛−^1 𝑒−𝑟^2 𝑟𝑑𝑟𝑑𝜙

∞ 0

𝜋/ 2 0 𝐿𝑎 𝑓𝑢𝑛𝑐𝑖ó𝑛 𝑒𝑛 𝑙𝑎 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑠𝑒 𝑝𝑢𝑒𝑑𝑒 𝑠𝑒𝑝𝑎𝑟𝑎𝑟 𝑒𝑛 𝑢𝑛 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑜, 𝑒𝑠𝑡𝑜 𝑒𝑠: 𝑓 𝑟, 𝜙 = 𝑔 𝑟 𝑕(𝜙) 𝑃𝑜𝑟 𝑡𝑎𝑛𝑜 𝑝𝑜𝑑𝑒𝑚𝑜𝑠 𝑠𝑒𝑝𝑎𝑟𝑎𝑟 𝑙𝑎 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑑𝑜𝑏𝑙𝑒 𝑐𝑜𝑚𝑜 𝑒𝑙 𝑝𝑟𝑢𝑐𝑡𝑜 𝑑𝑒 𝑑𝑜𝑠 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙𝑒𝑠 𝑠𝑖𝑚𝑝𝑙𝑒:

𝜋/ 2 0

∞ 0 𝑟

2𝑚+2𝑛−2𝑒−𝑟^2 𝑟𝑑𝑟

𝑢 = 𝑟^2

∞ 0

𝑟2 𝑚+𝑛−1^ 𝑒−𝑢^ 𝑑𝑢

∞ 0

Gamma Function and Beta Function Limits, Assignments of Mathematics

Related documents

Partial preview of the text

Download Gamma Function and Beta Function Limits and more Assignments Mathematics in PDF only on Docsity!

2 Γ 4 Γ 3 2)^

2 Γ 4 Γ 3 =^

2 3! 2! =^

2 Γ^

2 Γ^

2 +^1

2 Γ^

2 Γ^

2 +^1

2 Γ^

Γ 𝑛 = 𝑥𝑛−^1

Γ 1 / 2 = 𝑥^1 /^2 −^1

Γ 1 / 2 = 𝑥−^1 /^2

𝐻𝑎𝑐𝑖𝑒𝑛𝑑𝑜 𝑥 = 𝑢^2

Γ 1 / 2 = 𝑢−^1

𝑒−𝑢^2 2𝑢𝑑𝑢

Γ 1 / 2 = 2 𝑒−𝑢^2 𝑑𝑢

Γ 1 / 2 2 = 2 𝑒−𝑢^2 𝑑𝑢

2 𝑒−𝑣^2 𝑑𝑣

𝐼^2 𝑀 = 𝑒

𝐼^2 𝑀 = 𝑒−^ 𝑢^2 +𝑣^2 𝑑𝑢𝑑𝑣

𝑒−^ 𝑢^2 +𝑣^2 𝑑𝑢𝑑𝑣

≤ 𝐼^2 𝑀 ≤ 𝑒−^ 𝑢^2 +𝑣^2 𝑑𝑢𝑑𝑣

𝑅 2 = 𝑏^2 + 𝑐^2 = 𝑀^2 + 𝑀^2 = 2𝑀^2

𝑅 2 = 2 𝑀^2 = 2 𝑀

Γ 1 / 2 2 = 𝐼𝑀^2 = 4 𝑒−^ 𝑢^2 +𝑣^2

𝑥^2 𝑒−2𝑥^2 𝑑𝑥

Γ 𝑛 = 𝑥𝑛−^1

𝐵 𝑚, 𝑛 = 𝑥𝑚−1^ 1 − 𝑥 𝑛−1𝑑𝑥

𝐵 𝑚, 𝑛 = 𝑥𝑚−1^ 1 − 𝑥 𝑛−1𝑑𝑥

= 𝑦𝑛−1^ 1 − 𝑦 𝑚−1𝑑𝑦

𝐵 𝑚, 𝑛 = 𝑥𝑚−1^ 1 − 𝑥 𝑛−1𝑑𝑥

𝑥 = 1 𝜃 = 𝑎𝑟𝑐𝑠𝑒𝑛^ 1 =^ 𝑎𝑟𝑐𝑠𝑒𝑛1 =^ 𝜋/

𝐵 𝑚, 𝑛 = 𝑠𝑒𝑛^2 𝜃 𝑚−1^ 1 − 𝑠𝑒𝑛^2 𝜃 𝑛−

𝑐𝑜𝑠^2 𝜃

𝑫𝒆𝒎𝒐𝒔𝒕𝒓𝒂𝒄𝒊ó𝒏

𝑆𝑒𝑎 ∶ Γ 𝑚 = 𝑧𝑚−^1

𝐻𝑎𝑐𝑖𝑒𝑛𝑑𝑜: 𝑧 = 𝑥^2 𝑑𝑧 = 2𝑥𝑑𝑥

Γ 𝑚 = 𝑥^2 (𝑚−^1 )

𝑒−𝑥^2 2𝑥𝑑𝑥 = 2 𝑥2𝑚−^2 +^1

𝑒−𝑥^2 𝑑𝑥 = 2^ 𝑥2𝑚−^1

𝑒−𝑥^2 𝑑𝑥

Γ 𝑚 = 2 𝑥2𝑚−^1

𝑒−𝑥^2 𝑑𝑥

Γ 𝑛 = 2 𝑦2𝑛−^1

𝑒−𝑦^2 𝑑𝑦

Γ 𝑚 Γ 𝑛 = 2 𝑥2𝑚−^1

𝑒−𝑥^2 𝑑𝑥 2 𝑦2𝑛−^1

𝑒−𝑦^2 𝑑𝑦

Γ 𝑚 Γ 𝑛 = 4 𝑥2𝑚−^1 𝑦^2 𝑛−^1 𝑒−^ 𝑥^2 +𝑦^2 𝑑𝑥𝑑𝑦

Γ 𝑚 Γ 𝑛 = 4 𝑥2𝑚−^1 𝑦^2 𝑛−^1 𝑒−^ 𝑥^2 +𝑦^2 𝑑𝑥𝑑𝑦

Γ 𝑚 Γ 𝑛 = 4 𝑟𝑐𝑜𝑠𝜙 2𝑚−^1 𝑟𝑠𝑒𝑛𝜙 2 𝑛−^1 𝑒−𝑟^2 𝑟𝑑𝑟𝑑𝑑𝜙

Γ 𝑚 Γ 𝑛 = 4 𝑟𝑐𝑜𝑠𝜙 2𝑚−^1 𝑟𝑠𝑒𝑛𝜙 2 𝑛−^1 𝑒−𝑟^2 𝑟𝑑𝑟𝑑𝜙

2𝑚+2𝑛−2𝑒−𝑟^2 𝑟𝑑𝑟

𝑢 = 𝑟^2

𝑟2 𝑚+𝑛−1^ 𝑒−𝑢^ 𝑑𝑢

𝑢 𝑚+𝑛−1^ 𝑒−𝑢𝑑𝑢

Γ 𝑚 Γ 𝑛 = 𝐵 𝑚, 𝑛 Γ 𝑚 + 𝑛 𝐵 𝑚, 𝑛 = Γ^ 𝑚^ Γ^ 𝑛