Consider a function $f ( s )$ defined by the following integral for positive real numbers $s$.

$$f ( s ) = \int _ { 0 } ^ { \infty } t ^ { s - 1 } \exp ( - t ) \mathrm { d } t$$

Answer the following questions. You may answer without showing that the above integral converges.

(1) Find the value of $f ( 1 )$.

(2) The inequality $\exp ( t ) > \frac { t ^ { n } } { n ! }$ holds for any positive real number $t$ and non-negative integer $n$.

(a) For positive real numbers $s$, show the following inequality.

$$\int _ { 0 } ^ { 1 } t ^ { s - 1 } \exp ( - t ) \mathrm { d } t < \frac { 1 } { s }$$

(b) When $n > s > 0$, show that the following inequality holds for any real number $c$ that satisfies $c > 1$.

$$\int _ { 1 } ^ { c } t ^ { s - 1 } \exp ( - t ) \mathrm { d } t < \frac { n ! } { n - s }$$

(3) When the second-order derivative of $f ( s )$ is expressed as

$$\frac { \mathrm { d } ^ { 2 } f ( s ) } { \mathrm { d } s ^ { 2 } } = \int _ { 0 } ^ { \infty } g ( t , s ) \exp ( - t ) \mathrm { d } t$$

find a function $g ( t , s )$. You may answer without showing that the order of differentiation and integration can be exchanged.

(4) Find the value of $D$ defined as

$$D = \int _ { 0 } ^ { \infty } ( \log t ) ^ { 2 } \exp ( - t ) \mathrm { d } t - \left( \int _ { 0 } ^ { \infty } ( \log t ) \exp ( - t ) \mathrm { d } t \right) ^ { 2 }$$

Here, you may use the fact that the following relation holds.

$$\left. \frac { \mathrm { d } ^ { 2 } } { \mathrm {~d} s ^ { 2 } } \log f ( s ) \right| _ { s = 1 } = \frac { \pi ^ { 2 } } { 6 }$$

(5) Define a function $p ( r )$ for positive real numbers $r$ and $\alpha$ as

$$p ( r ) = \frac { r } { \alpha } \exp \left( - \frac { r ^ { 2 } } { 2 \alpha } \right)$$

Find the value of $S$ defined as

$$S = \int _ { 0 } ^ { \infty } ( \log r ) ^ { 2 } p ( r ) \mathrm { d } r - \left( \int _ { 0 } ^ { \infty } ( \log r ) p ( r ) \mathrm { d } r \right) ^ { 2 }$$