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EXERCISES (^127)
4.
Perform these integrations which involve the Dirac &function:
i. J_9dx 6(x - 1).
ii. 1: d x ( x 2 + 3)6(x - 5 ).
iii. 1: d x x 6 ( x 2 - 5 ).
iv. /: 5 dx x 6 ( x 2 + 5).
r 2 v v. d x ~ ( C O S X ).
vi. c’4 dx x2 ~ ( C O S X ).
(^10) d8(x - 5)
viii. /,,d. ( x 2 + 3) [ dx ]
Determine the integral properties of the “triplet” d26(t)/dt2 by evaluating the integrals
The function h(x) is generated from the function g ( x ) by the integral
If h ( x ) is the triangular pulse shown below, find and plot g ( x ).
128
- Given that
THE DIRAC 6-FUNCTION
find and plot the first and second derivatives of f ( x ).
- An ideal impulse moving in space and time can be described by the function,
f(x,t) = 106(x - vat),
where v, is a constant. Make a three-dimensional plot for f ( x , t ) vs. x and t to show how this impulse propagates. What are the dimensions of vo? How does the plot change if
where a, is a constant? What are the dimensions of a,?
7. Consider the sinc function sequence:
&(t) = sin (nn)/(rx).
(a) On the same graph, plot three of these functions with n = 1, n = 10, and
(b) Prove that the limit of the sinc sequence functions acts like a 6-function by
n = 100.
deriving the relation:
As a first step, try making the substitution y = nt. You will need to use the identity:
- The function G(cosx) can be written as a sum of Dirac 6-functions
n
Find the range for n and the values for the a, and the xn.
so that its charge density can be expressed as
- A single point charge qo is located at (1, 1,O) in a Cartesian coordinate system,
pc(x, y , z ) = qo 6(x - 1) w y - 1) &z).
