Complex Logarithms Explained
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Ammar AjmalComplex Logarithms Explained
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Ammar AjmalComplex Variables 3.
4 The Logarithmic Function
Real logarithm:
e log x x; x 0
Complex logarithm
But:
e log z z
e log z ik 2 e log z z; k 0,1,2,......
The logarithm of a number, even a positive real number, is multivalued. And for a
real positive only one value is real.
DEFINITION:
log z Log z i arg z, z 0
In polar variables:
Indeed:
log z Log r i , r 0
r z ; arg z
e log z e Logr i e Logr cos i sin rcos i sin z
The principal value of the logarithm z (Log z) the principal argument of z
Log z Log r i p
r z , p arg z, p
Complex Variables
All values of log z come from:
k 0,1,2,.....
log z Log r i p2
p 0,1,2,.....
log z Log r i p k 2
Because p l 2
EXAMPLE 1 Find Log(-1 - i) and find all values of log(-1 -i).
Solution. The complex number (-1 i):
Which gives:
2, p
3
4
So:
and
Complex Variables
EXAMPLE 2 Find Log(-10) and all values of log(-10).
Solution. The principal argument is +.
So:
Log(10) Log10 i 2.303 i
log( 10)
e Log10i k 2 e Log10 cos 2k i sin 2k 10
Check: e
sign
Identity
logx1 x2 log x1 log x2
logz1 z2 log z1 log z2
We always obtain one the possible log(z1z2) value:
log z1 Log r1 i 1 2n1
log z1 log z 2 Log r1 Log r2 i1 2 2n1 n2
log
z
Log
r
2
n
2
2
2
2
log z1 log z 2 Logr1r2 i1 2 2n1 n2
EXAMPLE:
z1 i; log z1 i 2
3
log z1 log z 2 i
2
z 2 1; log z 2 i
z1 z 2 i logz1 z 2 i 2
BUT :
i 2 2 i
Complex Variables
In addition:
logz1 z2 log z1 log z2 log z1 log z2 Logr1 r2 i1 2 2 n1 n2
Now:
log z n n log z ,
log z n m
In general:
n
log z
m
log e z ik 2
z
n any integer
m0
k 0,1,2,...
e z e x cos y i sin y ;
arg( e z ) y 2k ,
k 0,1,2...
EXAMPLE 3 Let z = 1 + 3ni. Find all values of log ez and state which one is the
same as z.
Solution.
e z e1i 3 ecos 3 i sin 3 e
log e1i 3 log e Log e i 2k
log e1i 3 1 i 2k k 1 log e1i 3 1 i3
4