Consider the gamma function defined by Γ(t)=∫0^∞ x^t-1 e^-x d x, t>0 . Prove the following: (a

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Consider the gamma function defined by Γ(t)=∫0^∞ x^t-1 e^-x...

Consider the gamma function defined by $$ \Gamma(t)=\int_0^{\infty} x^{t-1} e^{-x} d x, \quad t>0 . $$ Prove the following: (a) $\Gamma(t+1)=t \Gamma(t)$ for all $t>0$. (Since $\Gamma(1)=1$, this implies that $\Gamma(n+1)=n !, n=1,2, \cdots)$ (b) Show that $\Gamma(t)$ can be written as $$ \Gamma(t)=2^{1-t} \int_0^{\infty} z^{2 t-1} e^{-\frac{1}{2} z^2} d z, $$ for all $t>0$. (c) Using (b), we can write $$ \Gamma\left(\frac{1}{2}\right)=\sqrt{2} \int_0^{\infty} e^{-\frac{1}{2} z^2} d z, $$ and thus, $$ \begin{aligned} {\left[\Gamma\left(\frac{1}{2}\right)\right]^2 } & =2\left\{\int_0^{\infty} e^{-\frac{1}{2} x^2} d x\right\}\left\{\int_0^{\infty} e^{-\frac{1}{2} y^2} d y\right\} \\ & =2 \int_0^{\infty} \int_0^{\infty} e^{-\frac{1}{2}\left(x^2+y^2\right)} d x d y . \end{aligned} $$ Now use polar coordinates to evaluate the double integral and thereby show that $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$.

Consider the gamma function defined by
$$
\Gamma(t)=\int_0^{\infty} x^{t-1} e^{-x} d x, \quad t>0 .
$$
Prove the following:
(a) $\Gamma(t+1)=t \Gamma(t)$ for all $t>0$. (Since $\Gamma(1)=1$, this implies that $\Gamma(n+1)=n !, n=1,2, \cdots)$
(b) Show that $\Gamma(t)$ can be written as
$$
\Gamma(t)=2^{1-t} \int_0^{\infty} z^{2 t-1} e^{-\frac{1}{2} z^2} d z,
$$
for all $t>0$.
(c) Using (b), we can write
$$
\Gamma\left(\frac{1}{2}\right)=\sqrt{2} \int_0^{\infty} e^{-\frac{1}{2} z^2} d z,
$$
and thus,
$$
\begin{aligned}
{\left[\Gamma\left(\frac{1}{2}\right)\right]^2 } & =2\left\{\int_0^{\infty} e^{-\frac{1}{2} x^2} d x\right\}\left\{\int_0^{\infty} e^{-\frac{1}{2} y^2} d y\right\} \\
& =2 \int_0^{\infty} \int_0^{\infty} e^{-\frac{1}{2}\left(x^2+y^2\right)} d x d y .
\end{aligned}
$$
Now use polar coordinates to evaluate the double integral and thereby show that $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$.

Key Concepts

Gamma Function

The Gamma function is an extension of the factorial function to complex and real number arguments. It is defined by an improper integral that converges for positive values of the parameter. This function plays a pivotal role in various fields of mathematics, particularly in probability, statistics, and complex analysis, by providing a continuous analogue of the factorial function.

Recurrence Relation

A key property of the Gamma function is its recurrence relation, which states that ?(t+1) = t ?(t) for all t > 0. This relation can be established by applying integration by parts to the defining integral of the Gamma function. The recurrence property is instrumental in linking the Gamma function with the factorial function, especially when evaluating ?(n+1), where n is a natural number, yielding n!.

Change of Variables in Integration

Change of variables, or substitution, is a fundamental technique in integration that helps simplify integrals by transforming them into more manageable forms. In the context of the Gamma function, an appropriate substitution can recast the integral into a form that connects with known integrals, such as those involving exponential and power terms. This technique is essential when deriving alternate representations of the Gamma function.

Polar Coordinates in Double Integrals

The method of converting Cartesian coordinates into polar coordinates is a powerful tool for evaluating double integrals, especially when the integrand exhibits radial symmetry. In problems involving the square of the Gamma function, the transition to polar coordinates simplifies the computation by transforming the double integral into an easier-to-compute radial integral. This approach is crucial in proving results like ?(1/2) = ??.

Special Case - Gamma(1/2)

The evaluation of the Gamma function at 1/2 is a celebrated result in mathematics, where it is shown that ?(1/2) equals ??. This special case demonstrates an elegant connection between the Gamma function and the Gaussian integral, linking areas such as complex analysis, probability theory, and statistical mechanics. The result is often derived using properties of the Gamma function and techniques like completing the square and polar coordinate transformation.

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