24.1 Functions of a complex variable
students are more at ease with the former type of statement, despite its lack of
precision, whilst others, those who would contemplate only the latter, are usually
well able to supply it for themselves.
24.1 Functions of a complex variableThe quantityf(z) is said to be a function of the complex variablezif to every value
ofzin a certain domainR(a region of the Argand diagram) there corresponds
one or more values off(z). Stated like thisf(z) could be any function consisting
of a real and an imaginary part, each of which is, in general, itself a function ofx
andy. If we denote the real and imaginary parts off(z)byuandv, respectively,
then
f(z)=u(x, y)+iv(x, y).In this chapter, however, we will be primarily concerned with functions that are
single-valued, so that to each value ofzthere corresponds just one value off(z),
and are differentiable in a particular sense, which we now discuss.
A functionf(z) that is single-valued in some domainRisdifferentiableat thepointzinRif thederivative
f′(z) = lim
∆z→ 0[
f(z+∆z)−f(z)
∆z]
(24.1)exists and is unique, in that its value does not depend upon the direction in the
Argand diagram from which ∆ztends to zero.
Show that the functionf(z)=x^2 −y^2 +i 2 xyis differentiable for all values ofz.Considering the definition (24.1), and taking ∆z=∆x+i∆y, we have
f(z+∆z)−f(z)
∆z=(x+∆x)^2 −(y+∆y)^2 +2i(x+∆x)(y+∆y)−x^2 +y^2 − 2 ixy
∆x+i∆y=2 x∆x+(∆x)^2 − 2 y∆y−(∆y)^2 +2i(x∆y+y∆x+∆x∆y)
∆x+i∆y=2x+i 2 y+(∆x)^2 −(∆y)^2 +2i∆x∆y
∆x+i∆y.
Now, in whatever way ∆xand ∆yare allowed to tend to zero (e.g. taking ∆y=0and
letting ∆x→0 or vice versa), the last term on the RHS will tend to zero and the unique
limit 2x+i 2 ywill be obtained. Sincezwas arbitrary,f(z)withu=x^2 −y^2 andv=2xy
is differentiable at all points in the (finite) complex plane.
We note that the above working can be considerably reduced by recognisingthat, sincez=x+iy, we can writef(z)as
f(z)=x^2 −y^2 +2ixy=(x+iy)^2 =z^2.