Integral Evaluation - University of Houston

ECE 6382

Fall 2023 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

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download