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- 1.
- Consider the function f(x),
f(x) |
= |
![$\displaystyle \left\{\begin{array}{cc}
0 & \mbox{\hspace{1cm} $\mid x \mid > a/2$ } \\
1/a & \mbox{\hspace{1cm} $\mid x \mid \leq a/2$ }
\end{array} \right.$](img11.gif) |
(1) |
- (a)
- Plot f(x) for a = 1 and a = 1/2.
- (b)
- Evaluate
.
- 2.
- The Dirac delta-function
is defined as
![$\displaystyle \delta (x)$](img14.gif) |
= |
![$\displaystyle \lim_{a \rightarrow 0} f (x)$](img15.gif) |
(2) |
where f(x) is defined in problem
above. Obviously
is undefined. Evaluate
- (a)
-
.
- (b)
-
,
where b is a real positive constant.
- (c)
-
,
where b is a real positive constant.
- (d)
-
,
where b and c are real
positive constants.
- 3.
- Let g (x) be a smooth function, and `b' be a real constant. Show that
- (a)
-
![$\displaystyle \int_{- \infty}^{\infty} \delta (x - b) g(x) \,dx$](img21.gif) |
= |
g(b) |
(3) |
- (b)
-
![$\displaystyle \int_{- \infty}^{\infty} \delta (bx) g(x) \,dx$](img22.gif) |
= |
![$\displaystyle \frac{1}{\mid b \mid} g(0)$](img23.gif) |
(4) |
- 4.
- OPTIONAL : If g(x) is a smooth function with the zero
given by g(x0) = 0, show that
![$\displaystyle \delta (g(x))$](img24.gif) |
= |
![$\displaystyle \frac{\delta (x - x_0)}{\left\vert \frac{d}{dx} g(x_0)
\right\vert }$](img25.gif) |
(5) |
Using Eqn (
) show that
![$\displaystyle \delta (x^2 - a^2)$](img26.gif) |
= |
![$\displaystyle \frac{1}{2 \vert a\vert} [\delta (x - a) + \delta (x - a)]$](img27.gif) |
(6) |
©Vijay A. Singh
Next: About this document ...
Up: No Title
Previous: SET I: PRELIMINARY
Vivek Ranjan
2000-08-07