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PMAT 41073
ASSIGHNMENT
Student name: M.A.A.N Gunawardana
Student no: PS/2007/086
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Title Page No1. Question. 12. Answer for the part (a). 23.
Answer for the part (b). 34. Answer for the part (c). 95. Answer
for the part (d). 116. References 16
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Assignment
Question
Dirac delta function x is defined by
x = Show that (a)
(b) Where the integration includes the point x=a, and f(x) is
continuous at x=a.
Fourier series
converges uniformly to f(x) is
Show that Assuming parsevals equation
Deduce that
(c) Justify your answer for (c) using (b) Explain (c) and (d)
qualitatively.
Show also that the best constant that approximates f(x) is
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Answers:
a) Consider the Dirac delta function
(a). consider
dx
let gx=t
=
therefore
Since g x Therefore gt
Consider the following standard integral
= 1
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(b)..
Since f(x) is continuous at x=a and the integration includes the
point x=a we can integrate the
In the interval of- Consider Let Then f(a) can pull out from the
integration sign
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The above figure shows the graphical interpretation of Dirac
Delta function
Consider that f(x) is uniformly convergent to fourier series in
the interval (
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Consider
+
Consider
If m=0
=
Consider
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+
Since
=
=
*
+ Consider m=n,
(summation vanishes because we consider only m= n case)
{| } { }
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Consider
Consider the case m *
+
=
* , -
, -+Consider the case m = n
The summation sign vanishes
=
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= , - ,| -=
C). Consider the parsvals equation
Take any arbitrary n such that
Let be any non decreasing positive termed sequence s.t
is bounded above , and is increasing sequence hence is
convergent
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Similarly
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And
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d).
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. )/
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Consider that the Error of an approximation is E which given
by
Let us assume that our function is approximated by any arbitrary
constant
Reference
1. www.wikipedia/freeencyclopedia.com
2. PMAT 41073 Lecture note
3. The graphs were drawn by using Microsoft mathematics 4.0
software
