Page 409 - Schaum's Outline of Theory and Problems of Advanced Calculus

400 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16

16.3. Express each function in the form uðx; yÞþ ivðx; yÞ, where u and v are real:
3
3z
(a) z ; ðbÞ 1=ð1 zÞ; ðcÞ e ; ðdÞ ln z.
3
2
3
3
2
2
3
2
3
w ¼ z ¼ðx þ iyÞ ¼ x þ 3x ðiyÞþ 3xðiyÞ þðiyÞ ¼ x þ 3ix y 3xy iy 2
ðaÞ
3 2 2 3
¼ x 3xy þ ið3x y y Þ
3
2
3
2
Then uðx; yÞ¼ x 3xy ; vðx; yÞ¼ 3x y y .
1 1 1 1 x þ iy 1 x þ iy
1 z 1 ðx þ iyÞ 1 x iy 1 x þ iy ð1 xÞ þ y 2
ðbÞ w ¼ ¼ ¼ ¼ 2
1 x y
:
Then uðx; yÞ¼ 2 2 ; vðx; yÞ¼ 2 2
ð1 xÞ þ y ð1 xÞ þ y
3x
3x 3iy
3x
3z
e ¼ e 3ðxþiyÞ ¼ e e 3x and u ¼ e cos 3y; v ¼ e sin 3y
ðcÞ ¼ e ðcos 3y þ i sin 3yÞ
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i 1
2
2
ln z ¼ lnð e Þ¼ ln þ i ¼ ln x þ y þ i tan y=x and
ðdÞ
2
2
u ¼ lnðx þ y Þ; v ¼ tan 1 y=x
1
2
Note that ln z is a multiple-valued function (in this case it is infinitely many-valued), since can be
increased by any multiple of 2 . The principal value of the logarithm is defined as that value for which
0 @ < 2 and is called the principal branch of ln z.
16.4. Prove (a) sinðx þ iyÞ¼ sin x cosh y þ i cos x sinh y
(b) cosðx þ iyÞ¼ cos x cosh y i sin x sinh y.
ix
We use the relations e ¼ cos z þ i sin z; e ix ¼ cos z i sin z, from which
iz
iz
e e iz e þ e iz
;
sin z ¼ cos z ¼
2i 2
e iðxþiyÞ e iðxþiyÞ e ix y e ixþy
Then sin z ¼ sinðx þ iyÞ¼ ¼
2i 2i
y
y
1 y y e þ e y e e y
2i 2 2
¼ fe ðcos x þ i sin xÞ e ðcos x i sin xÞg ¼ ðsin xÞ þ iðcos xÞ
¼ sin x cosh y þ i cos x sinh y
e iðxþiyÞ þ e iðxþiyÞ
2
Similarly, cos z ¼ cosðx þ iyÞ¼
1
¼ fe ix y þ e ixþy 1 y y
2
2 g¼ fe ðcos x þ i sin xÞþ e ðcos x i sin xÞg
y y y y
e þ e e e
¼ cos x cosh y i sin x sinh y
¼ðcos xÞ iðsin xÞ
2 2
DERIVATIVES, CAUCHY-RIEMANN EQUATIONS
d
16.5. Prove that z z, where z is the conjugate of z, does not exist anywhere.
z
dz
d
By definition, f ðzÞ¼ lim f ðz þ zÞ f ðzÞ if this limit exists independent of the manner in which
dz z!0 z
z ¼ x þ i y approaches zero. Then