Back
to top
The complex multifunction i^z and its "branches"
Watch this thread
Announcements
Page 1 of 1
Go to first unread
Skip to page:
username1599987
Badges:
12
Rep:
?
You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#1
So I was looking at 
, obviously this is a multi-valued function (a multifunction) due to the fact that 
and 
is a multifunction. Now the usual multifunctions I know either have a finite or infinite number of branches at ALL points, except, of course, the singularities and the branch points.
For example:
depending on if we take 
respectively (or any other odd or even multiples of 
). And this applies to all the other complex numbers, except zero. And the same applies to the logarithm function, except here we have infinite solutions differing by an integer multiple of 
.
However, when we come to look at
, we find that not all points in 
map to multiple points in 
! Example: 
only, and 
. As we can see here not all points in the z-plane will have the same number of corresponding points in the w-plane. So how many branches does this multifunction have? And what are the branch points?
, obviously this is a multi-valued function (a multifunction) due to the fact that and is a multifunction. Now the usual multifunctions I know either have a finite or infinite number of branches at ALL points, except, of course, the singularities and the branch points.For example:
depending on if we take respectively (or any other odd or even multiples of ). And this applies to all the other complex numbers, except zero. And the same applies to the logarithm function, except here we have infinite solutions differing by an integer multiple of .However, when we come to look at
, we find that not all points in map to multiple points in ! Example: only, and . As we can see here not all points in the z-plane will have the same number of corresponding points in the w-plane. So how many branches does this multifunction have? And what are the branch points?
0
reply
Gregorius
Badges:
14
Rep:
?
You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#2
Report
#2
(Original post by gagafacea1)
So I was looking at, obviously this is a multi-valued function (a multifunction) due to the fact that and is a multifunction. Now the usual multifunctions I know either have a finite or infinite number of branches at ALL points, except, of course, the singularities and the branch points.
For example: depending on if we take respectively (or any other odd or even multiples of ). And this applies to all the other complex numbers, except zero. And the same applies to the logarithm function, except here we have infinite solutions differing by an integer multiple of .
However, when we come to look at, we find that not all points in map to multiple points in ! Example: only, and . As we can see here not all points in the z-plane will have the same number of corresponding points in the w-plane. So how many branches does this multifunction have? And what are the branch points?
So I was looking at
, obviously this is a multi-valued function (a multifunction) due to the fact that and is a multifunction. Now the usual multifunctions I know either have a finite or infinite number of branches at ALL points, except, of course, the singularities and the branch points.For example:
depending on if we take respectively (or any other odd or even multiples of ). And this applies to all the other complex numbers, except zero. And the same applies to the logarithm function, except here we have infinite solutions differing by an integer multiple of .However, when we come to look at
, we find that not all points in map to multiple points in ! Example: only, and . As we can see here not all points in the z-plane will have the same number of corresponding points in the w-plane. So how many branches does this multifunction have? And what are the branch points?
the answer depend upon whether is integer, rational or irrational.
0
reply
username1599987
Badges:
12
Rep:
?
You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#3
(Original post by Gregorius)
Take a look at the Wikipedia article on "Exponentiation" under the subheading "complex exponents with complex bases". For the answer depend upon whether is integer, rational or irrational.
Take a look at the Wikipedia article on "Exponentiation" under the subheading "complex exponents with complex bases". For
the answer depend upon whether is integer, rational or irrational.
0
reply
Gregorius
Badges:
14
Rep:
?
You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#4
Report
#4
(Original post by gagafacea1)
Hmmm so you're saying that this multifunction does not have any certain number of branches unless i restrict the domain to integer values? So would this be even considered a multifunction or a mapping for that matter?
Hmmm so you're saying that this multifunction does not have any certain number of branches unless i restrict the domain to integer values? So would this be even considered a multifunction or a mapping for that matter?
as you range over the branches of the logarithm, then it's horribly behaved.If you scroll down the page that I referenced to the section on "Failure of power and logarithm identities", you'll see some other problems that arise in this type of generalization.
1
reply
username1599987
Badges:
12
Rep:
?
You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#5
(Original post by Gregorius)
What this tells you is that generalizing certain well-known functional forms from the real numbers to the complex numbers is not without its difficulties! In this case, if you fix a branch of the logarithm, then all is well; but if you look at the set of possible values of as you range over the branches of the logarithm, then it's horribly behaved.
If you scroll down the page that I referenced to the section on "Failure of power and logarithm identities", you'll see some other problems that arise in this type of generalization.
What this tells you is that generalizing certain well-known functional forms from the real numbers to the complex numbers is not without its difficulties! In this case, if you fix a branch of the logarithm, then all is well; but if you look at the set of possible values of
as you range over the branches of the logarithm, then it's horribly behaved.If you scroll down the page that I referenced to the section on "Failure of power and logarithm identities", you'll see some other problems that arise in this type of generalization.
0
reply
X
Page 1 of 1
Go to first unread
Skip to page:
Quick Reply
