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Math. 4103, homework 7, solutions
- Find the Laurent expansion of the function
f (z) = (^) (z − 1)(zz − 3)
in the region 0 < |z − 1 | < 2.
By partial fractions and the geometric series
f (z) = −^1 /^2 z − 1
+ 3 /^2
z − 3
(z − 1)−^1 − 3 4
1 − z− 21
= − 12 (z − 1)−^1 +
∑^ ∞
n=
2 −n(z − 1)n,
which is the Laurent expansion of f.
- Expand the function f (z) = z^2 − 1 into the Taylor series centered at z = 3.
f (z) = ((z − 3) + 3)^2 − 1 = 8 + 3(z − 1) + (z − 1)^2.
- Expand the function f (z) = (^) (1−^1 z) 3 into the Taylor series centered at z = 0.
Notice that
f (z) =^12
d dz
(1 − z)−^1 =^12
d dz
n=
zn^ =^12
∑^ ∞
n=
n(n − 1)zn−^2
∑^ ∞
n=
(n + 2)(n + 1) 2
zn.
- Find the residue at 0 of the function
cot(z) z^4. 1
This function has a pole of order 5 at zero. Hence, the residue is equal to
d dz
z^5 cot(z) z^4
|z=0 =^1 4!
d dz
(z cot(z))|z=0.
Notice that, by Leibniz rule
( (^) d dz
(z cot(z)) =
( (^) d dz
((z sin−^1 (z))cos(z))
∑^4
k=
k!(4 − k)!
d dz
)k (z sin−^1 (z))
d dz
) 4 −k cos(z)
=cos′′′′(z) + 4(z sin−^1 (z))′cos′′′(z) + 6(z sin−^1 (z))′′cos′′(z) +4(z sin−^1 (z))′′′cos′(z) + (z sin−^1 (z))′′′′ =cos(z) − 4(z sin−^1 (z))′sin(z) − 6(z sin−^1 (z))′′cos(z) −4(z sin−^1 (z))′′′sin(z) + (z sin−^1 (z))′′′′.
Notice that
(4.3) (^) z sin−^1 (z) = z
n=
(−1)n (2n + 1)! z
2 n+
n=
(−1)n (2n + 1)! z
2 n
Let
f (u) =
n=
(−1)n (2n + 1)! u
n
Then
(4.4) z sin−^1 (z) = f (z^2 ) = 1 + a 2 z^2 + a 4 z^4 + ....
because lim z→ 0 sin z( z)= 1. Therefore,
(z sin−^1 (z))|z=0 = 1, (z sin−^1 (z))′|z=0 = 0, (z sin−^1 (z))′′|z=0 2 a 2 , (z sin−^1 (z))′′′|z=0 = 0, (z sin−^1 (z))′′′′|z=0 = 4!a 4.
By the chain rule
f ′(u) = −
n=
(−1)n (2n + 1)! u
n
n=
(−1)nn (2n + 1)! u
n− 1
C
e(z+z
dz =
C
ez^ ez−^1 dz =
C
∑^ ∞
n=
zn n! e
z−^1 dz
∑^ ∞
n=
n!
C
zn^ ez
− 1 dz =
∑^ ∞
n=
n! Res^0 (z
n (^) ez−^1 )
∑^ ∞
n=
n!(n + 1)!.
- Show that the singular point of the following function is a pole.
f (z) = e^2 z^ (z − 1)−^2.
Determine the order of the pole.
The singular point is z = 1. In fact
f (z) = e2(z−1)+2(z − 1)−^2 = e^2 e^2 z−1)(z − 1)−^2 =
∑^ ∞
n=
e^2 n^1! (z − 1)n−^2.
Hence, z = 1 is a pole of order 2.
- Give an example of a function f (z) which has an essential isolated singularity
at z = 3.
f (z) = e^1 /(z−3).
- Give an example of a function f (z) which has a non-isolated singularity at z = 3.
f (z) = (^) sin(1/^1 (z−3)).
- Compute the integral (^) ∫ (^) ∞
−∞
z^4 + 1
dz.
This was done in class. The answer is √π 2.
