Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
Di€erentiating this with respect to y, we obtain
@ 2 u
@y 2 ˆ@v @y
ˆ @u
@x
Finally, using Eq. (6.13), this becomes
…with the aid of the first of Eqs: …6:11††:
@ 2 u
@y 2 ˆu;
which, on substituting Eq. (6.15), becomes
e x 00 …y† ˆe x …y† or 00 …y† ˆ…y†:
This is a simple linear di€erential equation whose solution is of the form
Then
and
Therefore
…y† ˆA cos y ‡ B sin y:
u ˆ e x …y†
ˆ e x …A cos y ‡ B sin y†
v ˆ @u
@y ˆex …A sin y ‡ B cos y†:
e z ˆ u ‡ iv ˆ e x ‰…A cos y ‡ B sin y†‡i…A sin y B cos y†Š:
If this is to reduce to e x when y ˆ 0, according to (c), we must have
from which we ®nd
Finally we ®nd
e x ˆ e x …A iB†
A ˆ 1 and B ˆ 0:
e z ˆ e x‡iy ˆ e x …cos y ‡ i sin y†:
…6:16†
This expression meets our requirements (a), (b), and (c); hence we adopt it as the
de®nition of e z . It is analytic at each point in the entire z plane, so it is an entire
function. Moreover, it satis®es the relation
e z 1
e z2 ˆ e z 1‡z 2
: …6:17†
It is important to note that the right hand side of Eq. (6.16) is in standard polar
form with the modulus of e z given by e x and an argument by y:
mod e z je z jˆe x and arg e z ˆ y:
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