according to the definition of the gamma function 1 2 0 t 12 e t dt a show that the change of va

Name: according to the definition of the gamma function 1 2 0 t 12 e t dt a show that the change of variable t s 2 leads to 2 marks 1 2 2 0 e s 2 ds b because s in a is a dummy variable we can wri 11804
Uploaded: 2023-03-27T13:19:09-08:00
Duration: 5 min 2 s
Channel: Sri K
Description: according to the definition of the gamma function 1 2 0 t 12 e t dt a show that the change of variable t s 2 leads to 2 marks 1 2 2 0 e s 2 ds b because s in a is a dummy variable we can wri 11804

First, we need to make the change of variable t = s^2. This means that dt = 2s ds. Now we can re

Question

According to the definition of the Gamma function, Γ(1/2) = ∫₀∞ t^(-1/2) e^(-t) dt. (a) Show that the change of variable t = s^2 leads to [2 marks] Γ(1/2) = 2∫₀∞ e^(-s^2) ds. (b) Because s in (a) is a dummy variable, we can write Γ(1/2)^2 = 4 (∫₀∞ e^(-x^2) dx)(∫₀∞ e^(-y^2) dy) = 4 ∫₀∞ ∫₀∞ e^(-(x^2+y^2)) dxdy. Change this double integral to polar coordinates to perform the integration and conclude that Γ(1/2) = √π. [4 marks]

          According to the definition of the Gamma function,
Γ(1/2) = ∫₀∞ t^(-1/2) e^(-t) dt.
(a) Show that the change of variable t = s^2 leads to [2 marks]
Γ(1/2) = 2∫₀∞ e^(-s^2) ds. 
(b) Because s in (a) is a dummy variable, we can write
Γ(1/2)^2 = 4 (∫₀∞ e^(-x^2) dx)(∫₀∞ e^(-y^2) dy) = 4 ∫₀∞ ∫₀∞ e^(-(x^2+y^2)) dxdy.
Change this double integral to polar coordinates to perform the integration and conclude that Γ(1/2) = √π. [4 marks]

Added by Kara C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

03/27/2023

Step 1

This means that dt = 2s ds. Now we can rewrite the integral: Γ(1/2) = ∫₀^∞ t^(-1/2) e^(-t) dt = ∫₀^∞ (s^2)^(-1/2) e^(-s^2) 2s ds = 2 ∫₀^∞ e^(-s^2) ds Now, let's square Γ(1/2) and rewrite it using the dummy variables x and y: Γ(1/2)^2 = (2 ∫₀^∞ e^(-x^2) dx)(2 Show more…

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According to the definition of the Gamma function, Γ(1/2) = ∫₀∞ t^(-1/2) e^(-t) dt. (a) Show that the change of variable t = s^2 leads to [2 marks] Γ(1/2) = 2∫₀∞ e^(-s^2) ds. (b) Because s in (a) is a dummy variable, we can write Γ(1/2)^2 = 4 (∫₀∞ e^(-x^2) dx)(∫₀∞ e^(-y^2) dy) = 4 ∫₀∞ ∫₀∞ e^(-(x^2+y^2)) dxdy. Change this double integral to polar coordinates to perform the integration and conclude that Γ(1/2) = √π. [4 marks]