Page 43 - Calculus with Complex Numbers

W e get the Laurent expansion at z = -ï by putting t = z + i and expanding in
terms of t. This time we have

1 1 1 t
1 = - + + + . . . y
+ c2 1it k qk
which shows that f (z) also has a simple pole at z = -ï but now with
residue - 1/2ï .

N otes
For a proof of Theorem 2 see for example Knopp (1945) page 30.
Neither Taylor nor M aclaurin gave a rigorous proof of the validity of their
expansions. They are not valid in general even for functions with derivatives of
all orders. An interesting example is the function
flx) = e 1/x2
-
,
which (if we assume f @) = 0 at .x = 0) has fçns (0) = 0 for all n, so has
a Maclaurin expansion which vanishes identically, therefore cannot = flx) at
any .x # 0.
They are of course valid for the elementary functions we corlsider here.
Rigorous treatments of complex analysis are able to give proofs of the validity
of Taylor M aclaurin and Laurent expansiorls in the complex domain using the
theol'y of contour integration developed in the next chapter. (See Knopp (1945)
chapter 7 for the details.)

Exam ples

Verify the Cauchy-lkiemann equations for the following functions:

sinz, log z.
Verify the Cauchy-lkiemann formula for the derivative in each case.

Prove Iz I2 is differentiable only at z = 0. What is its derivative at this point?
Prove flz) = i (1zl2 - 2) is differentiable only on the unit circle Iz I = 1.

Verify that f' (z) = 12 for these z.
Prove that if f (z) is differentiable for all z and is evelywhere real valued then
flz) must be corlstant.
Find the Maclaurin exparlsion of ez sin z up to terms in .5 (i) by differentiating
z
and putting z = 0, (ii) by multiplying the Maclaurin expansiorls of ez and sin z
together.
Find the Taylor expansiorls of the following functions at the points indicated.
State the range of validity in each case.
!
(i) 1/z at .: = 2, (ii) ez at .!: = i, (iii) log .! (PV) at .: = 1.
!
: