1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2.1 • FUNCTIONS AND L INEAR MAPPINGS 51

Examples 2.1 and 2.2 show how to find u(x, y) and v(x, y) when a rule
for computing f is given. Conversely, if u(x, y) and v (x, y) are two real-valued
functions of the real variables x and y, they determine a complex-valued function
f (x , y) = u (x, y) +iv (x, y), and we can use the formulas
z + z z-z
X = -- and y=--
2 2i
to find a formula for f involving the variables z and z.

• EXAMPLE 2.3 Express f (z) = 4x^2 + i4y^2 by a formula involving the vari-
  ables z and :z.

Solution Calculation reveals that

f (z) = 4 ( z; :Zr+ i4 ( z ~ z) 2

= z^2 + 2zz + z^2 - i (z^2 - 2zz + :z^2 )

= (1 - i) z^2 + (2 + 2i) zz + (1 - i) z^2 •

Using z = r ei^9 in the expression of a complex function f may be convenient.

It gives us the polar representation

f(z) = f.(rei^8 ) =u(r, O)+iv(r, 0), (2-2)

where u and v are real functions of the real variables rand 0.

Remark 2.1 For a given function /, the functions u and v defined h ere are
different from those defined by Equation (2-1) because Equation (2-1) involves
Cartesian coordinates and Equation (2-2) involves polar coordinates. •

• EXAMPLE 2.4 Express f (z) = z^2 in both Cartesian and polar form.

Solution For the Cartesian form, a simple calculation gives

f (z) = f (x+iy) = (x+iy)

2

= (x^2 - y^2 ) + i(2xy) =u(x, y) +iv(x, y)

so that

u(x, y) = x^2 -y^2 , and v(x, y) = 2xy.

For the polar form, we refer to Equation (1-39) to get

f (rei^8 ) = (rei^8 )

2

= r^2 ei^28 = r^2 cos 28 + ir^2 sin 20 = U (r, 0) + iV (r, 8),

so that

U(r,O)=r^2 cos28, and V(r,8)=r^2 sin28.