# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Wed, 19 Jun 2013 23:32:01 UTC

 All times are UTC [ DST ]

 Page 1 of 3 [ 36 posts ] Go to page 1, 2, 3  Next
 Print view Previous topic | Next topic
Author Message
 Post subject: Entire FunctionsPosted: Sun, 19 Feb 2012 22:19:36 UTC
 Member

Joined: Thu, 26 Jan 2012 05:30:38 UTC
Posts: 25
Show there are two entire complex functions f1 and f2 that satisfy
where
f1(0)=1
f1'(0)=0
f2(0)=0
f2'(0)=1

Assume the solution has the following form

The second derivative = zf(z) so we get

a_0=2*3a_3
a_1=3*4a_4
a_2=4*5a_5 ...
a_n = a_(n-3) / (n-1)n

How do I get a_2?

Top

 Post subject: Re: Entire FunctionsPosted: Sun, 19 Feb 2012 22:42:49 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12172
Location: Austin, TX
monomoco wrote:
Show there are two entire complex functions f1 and f2 that satisfy
where
f1(0)=1
f1'(0)=0
f2(0)=0
f2'(0)=1

Assume the solution has the following form

The second derivative = zf(z) so we get

a_0=2*3a_3
a_1=3*4a_4
a_2=4*5a_5 ...
a_n = a_(n-3) / (n-1)n

How do I get a_2?

Where are and ? All I see is one .

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Entire FunctionsPosted: Sun, 19 Feb 2012 22:59:50 UTC
 Member

Joined: Thu, 26 Jan 2012 05:30:38 UTC
Posts: 25
That's what I'm trying to find, the two functions f1 and f2.

Top

 Post subject: Re: Entire FunctionsPosted: Sun, 19 Feb 2012 23:08:59 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12172
Location: Austin, TX
monomoco wrote:
That's what I'm trying to find, the two functions f1 and f2.

OK, then here they are:

and

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Entire FunctionsPosted: Sun, 19 Feb 2012 23:25:44 UTC
 Member

Joined: Thu, 26 Jan 2012 05:30:38 UTC
Posts: 25
I don't think that's what I was supposed to find... My prompt tells me I'm supposed to find a recurrence relation for the coefficients a_n and then show the resulting series have an infinite radius of convergence using the ratio test.

Top

 Post subject: Re: Entire FunctionsPosted: Sun, 19 Feb 2012 23:42:35 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12172
Location: Austin, TX
monomoco wrote:
I don't think that's what I was supposed to find... My prompt tells me I'm supposed to find a recurrence relation for the coefficients a_n and then show the resulting series have an infinite radius of convergence using the ratio test.

Not sure what to tell you, from what you posted the I have provided are just fine. Again, I think you should reread what you originally posted and check to make sure you don't have any typos.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Entire FunctionsPosted: Sun, 19 Feb 2012 23:47:14 UTC
 Member

Joined: Thu, 26 Jan 2012 05:30:38 UTC
Posts: 25
There are no typos. How does f(z)=1 satisfy f''=zf(z)? Or f(z)=z?

Top

 Post subject: Re: Entire FunctionsPosted: Sun, 19 Feb 2012 23:52:04 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12172
Location: Austin, TX
monomoco wrote:
There are no typos. How does f(z)=1 satisfy f''=zf(z)? Or f(z)=z?

It doesn't, but you didn't say , they're different function names. If you intend them to be the same function, you should write it as and not

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Entire FunctionsPosted: Sun, 19 Feb 2012 23:59:43 UTC
 Member

Joined: Thu, 26 Jan 2012 05:30:38 UTC
Posts: 25
Sorry it was not clear. I am looking for f1(z) and f2(z)that satisfy the DE and initial conditions and have that form of the series.

Top

 Post subject: Re: Entire FunctionsPosted: Mon, 20 Feb 2012 00:04:33 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12172
Location: Austin, TX
monomoco wrote:
Sorry it was not clear. I am looking for f1(z) and f2(z)that satisfy the DE and initial conditions and have that form of the series.

Both of them need to satisfy the same differential equation or one of them goes on the left side and the other on the right side?

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Entire FunctionsPosted: Mon, 20 Feb 2012 00:07:32 UTC
 Member

Joined: Thu, 26 Jan 2012 05:30:38 UTC
Posts: 25
They both need to satisfy the same DE

Top

 Post subject: Re: Entire FunctionsPosted: Mon, 20 Feb 2012 00:18:27 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12172
Location: Austin, TX
monomoco wrote:
They both need to satisfy the same DE

OK, now that makes a lot more sense.

So you have

Expanding , we get:

Can you see where you go from here?

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Entire FunctionsPosted: Mon, 20 Feb 2012 00:28:21 UTC
 Member

Joined: Thu, 26 Jan 2012 05:30:38 UTC
Posts: 25
I don't understand how you expanded that. I began listing the terms in the series

f''= 2a_2 + z*6a_3 + z^2*12a_4 + ... and set it to
zf(z)=z*a_0 + z^2*a_1 + z^3*a_2+...

and started looking at the coefficients of z^i

For z : 6a_3 = a_0
For z^2: 12a_4 = a_1

etc.

Top

 Post subject: Re: Entire FunctionsPosted: Mon, 20 Feb 2012 00:32:26 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12172
Location: Austin, TX
monomoco wrote:
I don't understand how you expanded that. I began listing the terms in the series

f''= 2a_2 + z*6a_3 + z^2*12a_4 + ... and set it to
zf(z)=z*a_0 + z^2*a_1 + z^3*a_2+...

and started looking at the coefficients of z^i

For z : 6a_3 = a_0
For z^2: 12a_4 = a_1

etc.

I did the same thing as you did, I just did it all at once using sigma-notation rather than write individual terms. I mean, one already knows what the are by Taylor's theorem, so may as well use it.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Entire FunctionsPosted: Mon, 20 Feb 2012 00:38:04 UTC
 Member

Joined: Thu, 26 Jan 2012 05:30:38 UTC
Posts: 25
OK, so this is where I was stuck when I first posted. I don't know where to go from here.

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 3 [ 36 posts ] Go to page 1, 2, 3  Next

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous