Integral Evaluation - University of Houston
ECE 6382
Fall 2022 David R. Jackson
Notes 11 Evaluation of Definite Integrals
via the Residue Theorem
Notes are from D. R. Wilton, Dept. of ECE
1
Review of Singular Integrals
Ln x
1
x
Ln= ( z) ln (r) + i
- < <
Logarithmic singularities are examples of integrable singularities:
( ) 1 Ln
0
(
x)
dx
1
= lim0
Ln
(
x
)
dx
= lim xLn ( x) - 0
x
1 x=
= -1 since
lim xLn ( x) = 0
x0
Note: There might be numerical trouble if one integrates this function numerically!
2
Review of Singular Integrals (cont.)
Singularities like 1/x are non-integrable.
1 x
x
1
( ) ( ) 1 1 dx =
0x
lim
0
1 1 dx = x
lim
0
Ln
x
1
= -
x=
3
Review of Cauchy Principal Value Integrals
Consider the following integral:
1/ x
I =
2 dx = -1 x
0 dx + -1 x
2 dx 0x
=
Ln
x
0 x
= -1 + Ln x
2 x
= 0
-1
-
x
2
A finite result is obtained if the integral interpreted as
Excluded region
( ) I =
2 dx = lim
-1 x
0
- dx + -1 x
2 dx x
=
lim
0
Ln
x
-x= -1 + Ln x
2
x= +
( ) = lim Ln - Ln1 + Ln2 - Ln = Ln2
0
The infinite contributions from the two symmetrical shaded parts shown exactly cancel in this limit. Integrals evaluated in this way are said to be (Cauchy) principal value (PV) integrals:
Not= ation: I P= V 2 dx or 2 dx
-1 x
-1 x
4
Cauchy Principal Value Integrals (cont.)
1/x singularities are examples of singularities integrable only in the principal value (PV) sense.
Principal value integrals must not start or end at the singularity, but must pass through them to permit cancellation of infinities
0
1/ x
a -
x
b
Excluded region
5
Cauchy Principal Value Integrals (cont.)
1/ x2
Singularities like 1/x2 are non- integrable (even in the PV sense).
x a - b
- 1
a x2
dx +
b1 x2
d=x
1 a
+
1
+
1
-
1 b
but note that
sgn( x) x2
=
-
1 x2 1 x2
, ,
x>0 x ................
................
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