Page 127 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

COMPLEX VARIABLES AND THE LAPLACE INVERSION INTEGRAL 117
y
1 B

A
0
x
1 2

FIGURE 7.5: Integration of an analytic function along two paths

in two dimensions. We mention here that it is also important in the study of incompressible,
inviscid fluid mechanics and in other areas of science and engineering. You will undoubtedly
meet with it in some of you clurses.

Example 7.4.

f = z 2 f = 2z

f = sin z f = cos z
f = e az f = ae az

Integrals
Consider the line integral along a curve C defined as x = 2y from the origin to the point
x = 2, y = 1, path OB in Fig. 7.5.

2
z dz
C

We can write

2
2
2
2
2
z = x − y + 2ixy = 3y + 4y i
and dz = (2 + i)dy
Thus
1 1
2 11

2
2
2
(3y + 4y i)(2 + i)dy = (3 + 4i)(2 + i) y dy = + i
3 3
y=0 y=0