Gamma function The gamma function is defined by T(p) = ™ 2P a. Use the reduction formula 1. x²e-¹ dx dx, for p not equal to zero or a negative integer. 1² e-4² du = √.00 P to show that I'(p+1) =p! (p factorial). b. Use the substitution x = u² and the fact that √T 2 xPle dx for p = 1, 2, 3, ... to show that I 2 = √T.
Gamma function The gamma function is defined by T(p) = ™ 2P a. Use the reduction formula 1. x²e-¹ dx dx, for p not equal to zero or a negative integer. 1² e-4² du = √.00 P to show that I'(p+1) =p! (p factorial). b. Use the substitution x = u² and the fact that √T 2 xPle dx for p = 1, 2, 3, ... to show that I 2 = √T.
Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Gamma Function**: The gamma function is defined by
\[
\Gamma(p) = \int_{0}^{\infty} x^{p-1} e^{-x} \, dx,
\]
for \( p \) not equal to zero or a negative integer.
a. Use the reduction formula
\[
\int_{0}^{\infty} x^{p} e^{-x} \, dx = p \int_{0}^{\infty} x^{p-1} e^{-x} \, dx \quad \text{for } p = 1, 2, 3, \ldots
\]
to show that \(\Gamma(p + 1) = p! \) (p factorial).
b. Use the substitution \( x = u^2 \) and the fact that
\[
\int_{0}^{\infty} e^{-u^2} \, du = \frac{\sqrt{\pi}}{2}
\]
to show that \(\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25b167f9-3118-479d-a9fb-f8c446c41f1a%2F8e8f54aa-935c-4d25-ac0f-7f91df3139ca%2Fhok3pxi_processed.png&w=3840&q=75)
Transcribed Image Text:**Gamma Function**: The gamma function is defined by
\[
\Gamma(p) = \int_{0}^{\infty} x^{p-1} e^{-x} \, dx,
\]
for \( p \) not equal to zero or a negative integer.
a. Use the reduction formula
\[
\int_{0}^{\infty} x^{p} e^{-x} \, dx = p \int_{0}^{\infty} x^{p-1} e^{-x} \, dx \quad \text{for } p = 1, 2, 3, \ldots
\]
to show that \(\Gamma(p + 1) = p! \) (p factorial).
b. Use the substitution \( x = u^2 \) and the fact that
\[
\int_{0}^{\infty} e^{-u^2} \, du = \frac{\sqrt{\pi}}{2}
\]
to show that \(\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}\).
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