Cauchy-Riemann equations
Carlo Teubner
If at a point x + iy the Cauchy-Riemann equations hold, and u and v are
continuous, then f is holomorphic at x + iy.
Now my question is this: If the Cauchy-Riemann equations hold in an open
subset U of C, does this already imply that f is holomorphic in U - i.e.
without the extra assumption of continuity of u, v.
I couldn't find anything about this anywhere; except that Wikipedia
states it, but without proof
(http://en.wikipedia.org/wiki/Cauchy-Riemann_equations).
I'd be grateful for elucidation.
Carlo
--
C Teubner t offline de
dot at minus dot
...oops: "on" not "off" :)
c_hi...@gmx.de
I think there's a counterexample in my textbook.
Take the function f(z)=e^(-1/(z^4)) for z!=0 and f(0)=0.
The equations hold for all z in C, but the function is not
differentiable in the origin.
David C. Ullrich
signature."@example.com> wrote:
>Let f: C -> C, f = u + iv.
>
>If at a point x + iy the Cauchy-Riemann equations hold, and u and v are
>continuous, then f is holomorphic at x + iy.
No. Well, actually this doesn't quite even make sense: One
speaks of a function being holomorphic in an open set.
But the hypotheses you give do not imply that f is
differentiable at x + iy.
It's true if the first-order partial derivatives of u and v
are continuous at x + iy.
>Now my question is this: If the Cauchy-Riemann equations hold in an open
>subset U of C, does this already imply that f is holomorphic in U - i.e.
>without the extra assumption of continuity of u, v.
I believe that this is true, although I could be confusing it
with something else. I think it's a theorem of Hartogs, which
you can find in Narasimhan's book on one complex variable.
>I couldn't find anything about this anywhere; except that Wikipedia
>states it, but without proof
>(http://en.wikipedia.org/wiki/Cauchy-Riemann_equations).
>
>I'd be grateful for elucidation.
>
>Carlo
************************
David C. Ullrich
David C. Ullrich
Oops. I should have read this before posting my reply...
************************
David C. Ullrich
KRamsay
In article <cvv2t09k03dnvianv...@4ax.com>, David C. Ullrich
<ull...@math.okstate.edu> writes:
>I think it's a theorem of Hartogs, which
>you can find in Narasimhan's book on one complex variable.
The two theorems of Hartogs mentioned in the index of that
book are in the chapter on several complex variables.
The most relevant theorem in the book appears to be the
Looman-Menchoff theorem, which Carlo Teubner already
alluded to: if Omega is an open set, f is continuous and
has both partial derivatives on Omega, and f satisfies the
Cauchy-Riemann equations on Omega, then f is holomorphic
on Omega. At the expense of a little bit more delicate proof,
this slightly strengthens the easier result that also assumes
the partial derivatives of f are continuous. And now we see
that the continuity of f can't be removed.
Keith Ramsay
P.S. It was good that we had that book. In the first week I
had taken more than twice the notes I had taken per week
in most of my other math courses at that level, and I had to
recopy them for legibility. But then I noticed how closely he
was following his book and it got much easier. And of course
it's a good book anyway.
One day Jesse Jackson came to campus and was giving a
speech during Narasimhan's class. From in front of me I was
getting Narasimhan with his audible but quiet, precise, and
quick voice; from behind, outside, I was getting THUNDERING
appeals to END the RACISM of apartheid. Hallelujah! Gather
unto me those sheaves of holomorphic functions!
David C. Ullrich
>
>In article <cvv2t09k03dnvianv...@4ax.com>, David C. Ullrich
><ull...@math.okstate.edu> writes:
>>I think it's a theorem of Hartogs, which
>>you can find in Narasimhan's book on one complex variable.
>
>The two theorems of Hartogs mentioned in the index of that
>book are in the chapter on several complex variables.
>
>The most relevant theorem in the book appears to be the
>Looman-Menchoff theorem, which Carlo Teubner already
>alluded to: if Omega is an open set, f is continuous and
>has both partial derivatives on Omega, and f satisfies the
>Cauchy-Riemann equations on Omega, then f is holomorphic
>on Omega.
Yes, that's the theorem I was thinking of.
Just because I'm getting more or less everything wrong here
I feel the need to point out that this is not the result
that Carlo stated. I actually was right when I said that
his statement
"If at a point x + iy the Cauchy-Riemann equations hold,
and u and v are continuous, then f is holomorphic at x + iy."
was false (assuming we take "holomorphic at x + iy" to mean
"differentiable at x + iy".) For example let v = 0, let
u(0) = 0 and u(r exp(it)) = r sin(4t). Everything's continuous,
the Cauchy-Riemann equations hold at the origin, but f is
not differentiable at the origin.
>At the expense of a little bit more delicate proof,
>this slightly strengthens the easier result that also assumes
>the partial derivatives of f are continuous. And now we see
>that the continuity of f can't be removed.
>
>Keith Ramsay
>
>P.S. It was good that we had that book. In the first week I
>had taken more than twice the notes I had taken per week
>in most of my other math courses at that level, and I had to
>recopy them for legibility. But then I noticed how closely he
>was following his book and it got much easier. And of course
>it's a good book anyway.
>
>One day Jesse Jackson came to campus and was giving a
>speech during Narasimhan's class. From in front of me I was
>getting Narasimhan with his audible but quiet, precise, and
>quick voice; from behind, outside, I was getting THUNDERING
>appeals to END the RACISM of apartheid. Hallelujah! Gather
>unto me those sheaves of holomorphic functions!
************************
David C. Ullrich
KRamsay
<ull...@math.okstate.edu> writes:
|On 28 Dec 2004 22:09:41 GMT, kra...@aol.com (KRamsay) wrote:
|>In article <cvv2t09k03dnvianv...@4ax.com>, David C. Ullrich
|><ull...@math.okstate.edu> writes:
|>>I think it's a theorem of Hartogs, which
|>>you can find in Narasimhan's book on one complex variable.
|>
|>The two theorems of Hartogs mentioned in the index of that
|>book are in the chapter on several complex variables.
|>
|>The most relevant theorem in the book appears to be the
|>Looman-Menchoff theorem, which Carlo Teubner already
|>alluded to: if Omega is an open set, f is continuous and
|>has both partial derivatives on Omega, and f satisfies the
|>Cauchy-Riemann equations on Omega, then f is holomorphic
|>on Omega.
|
|Yes, that's the theorem I was thinking of.
|
|Just because I'm getting more or less everything wrong here
|I feel the need to point out that this is not the result
|that Carlo stated.
This is true.
When I was composing my last posting, I had thought in all
honesty that I was covering myself by switching from "described"
to "alluded to". I guess that's a little lame. He had quoted a
web page that I assumed had the correct statement.
Actually, though, the version of the theorem given on the Wikipedia
page is a relatively weak one. It includes the hypothesis that
Re(f) and Im(f) are differentiable considered as functions of two
real variables, which is stronger than their merely having partial
derivatives in the region (and implies that they are continuous).
Assuming that makes the transition to "complex differentiable"
relatively small.
Incidentally, the definition of "holomorphic at a point" that
I remember from school is that f is said to be holomorphic at
a point if f is holomorphic on an open neighborhood of the point.
Keith Ramsay
David C. Ullrich
Good for Wikipedia!
Assuming differentiability in this sense doesn't give the
best results of the sort we're talking about here, but
it _is_ the way to understand _pointwise_ differentiability.
Let's say f is R-differentiable if it's differentiable as
a map from R^2 to R^2 (there exists a linear map etc)
and C-differentiable if it's differentiable in the sense
of complex analysis. Then f is C-differentiable at a point
if and only if it's R-differentiable and the C-R equations
hold (and this is equivalent to "f is R-differentiable,
and the R-derivative, which is a priori an R-linear
map from R^2 to R^2, is actually C-linear as a map
from C to C".)
So R-differentiability is the way to look at it,
because (i) it's the only way to give a _pointwise_
"f is C-differentiable at z if and only if ..."
where "..." has something to do with the C-R equations
and (ii) it explains what the C-R equations "really
mean": they mean just that the R-derivative is
C-linear.
>Incidentally, the definition of "holomorphic at a point" that
>I remember from school is that f is said to be holomorphic at
>a point if f is holomorphic on an open neighborhood of the point.
Are you sure you've seen this in a standard reference?
I haven't - all I'm familiar with is that f is _analytic_
at a point if it's holomorphic in a neighborhood of the
point.
>Keith Ramsay
************************
David C. Ullrich
Carlo Teubner
problem sheet rather than filter it through my uneducated and
error-prone undergraduate mind.
"Show that if f is a differentiable function satisfying df/dz-bar = 0 on
D then f is holomorphic on D." [D being an open disc]
df/dz-bar is given as 1/2*(df/dx + i df/dy).
It's easy to derive the Cauchy-Riemann equations from df/dz-bar = 0.
One problem with the question is that it's not clear to me what
"differentiable" means, though from context I might guess it means
"differentiable as a function of x and y" (since that has been mentioned
before). Or maybe it means that df/dz and df/dz-bar exist, who knows. I
doubt it has to do with R-differentiability, which you mentioned, which
we haven't come across yet.
Anyway, thanks so far to both of you for your help.
>>Incidentally, the definition of "holomorphic at a point" that
>>I remember from school is that f is said to be holomorphic at
>>a point if f is holomorphic on an open neighborhood of the point.
This is the definition we've learned.
Dave L. Renfro
[sci.math Dec 28 2004 5:35:03:000PM]
http://mathforum.org/discuss/sci.math/m/666165/666275
wrote (in part):
> The most relevant theorem in the book appears to be the
> Looman-Menchoff theorem, which Carlo Teubner already
> alluded to: if Omega is an open set, f is continuous and
> has both partial derivatives on Omega, and f satisfies the
> Cauchy-Riemann equations on Omega, then f is holomorphic
> on Omega. At the expense of a little bit more delicate proof,
> this slightly strengthens the easier result that also assumes
> the partial derivatives of f are continuous. And now we see
> that the continuity of f can't be removed.
In fact, Looman and Menchoff [= Men'shov] actually showed
that we only need the partials to exist finitely on a
co-countable set. Sindalovskii [3] observed (top of p. 360)
that it follows from Robert M. Fesq (MR 30 #4896) and Paul
J. Cohen (MR 21 #3004) [yes, the Cohen who proved the axiom
of choice and continuum hypothesis independence results]
that we can strengthen this even further. Instead of the
partials existing finitely on all but a countable set,
it is enough to assume that the partials exist finitely
on all but a subset of a countable union of closed sets
each having finite Hausdorff 1-measure, or equivalently,
as long as the partials exist everywhere except on a set
that can be covered by an F_sigma set with sigma-finite
Hausdorff 1-measure. Incidentally, this is a strictly
stronger notion of smallness than being simultaneously
meager (i.e. first category) in R^2 and having sigma-finite
Hausdorff 1-measure -- see Remark 1 in Renfro [3], where
in (a) I neglected to include the hypothesis that the
set is also meager.
By the way, I know next to nothing about the theory
behind these results. I just happen to be interested
in various hierarchies of "small set notions". When
I come across a paper that gives a non-trivial use of
some notion of smallness of interest to me, I often
write a note to myself about it or file a copy of the
paper away somewhere. So, when I saw these posts about
Cauchy-Riemann weakenings, I remembered having a paper
on this topic that involved a Hausdorff measure version
of F_sigma measure zero sets, and hence, I knew it would
be in my notebooks that deal with subsets of F_sigma
measure zero sets.
[1] Shaun A. R. Disney, Jack D. Gray, and Sidney A. Morris,
"Is a function that satisfies the Cauchy-Riemann
equations necessarily analytic?", The Australian
Mathematical Society Gazette 2(3) (1975), 67-81.
[2] Jack D. Gray and Sidney A. Morris, "When is a function
that satisfies the Cauchy-Riemann equations analytic?",
The American Mathematical Monthly 85 (1978), 246-256.
[3] Dave L. Renfro, 1 May 2000 sci.math post "HISTORICAL
ESSAY ON F_SIGMA LEBESGUE NULL SETS".
http://mathforum.org/discuss/sci.math/t/267778
[4] G. Kh. Sindalovskii, "Cauchy-Riemann conditions
in a class of functions with summable modulus, and
certain boundary properties of analytic functions",
Math. USSR Sbornik 56 (1987), 359-377.
Dave L. Renfro
David C. Ullrich
signature."@example.com> wrote:
>Ok, it's probably best if I transcribe the question verbatim from my
>problem sheet rather than filter it through my uneducated and
>error-prone undergraduate mind.
>
>"Show that if f is a differentiable function satisfying df/dz-bar = 0 on
>D then f is holomorphic on D." [D being an open disc]
That's very different from your original
"If at a point x + iy the Cauchy-Riemann equations hold,
and u and v are continuous, then f is holomorphic at x + iy."
Big differences are that f is supposed to be "differentiable",
not just continuous, and also the Cauchy-Riemann equations
are given to hold in an entire open set, not just at a point.
>df/dz-bar is given as 1/2*(df/dx + i df/dy).
>
>It's easy to derive the Cauchy-Riemann equations from df/dz-bar = 0.
>
>One problem with the question is that it's not clear to me what
>"differentiable" means, though from context I might guess it means
>"differentiable as a function of x and y" (since that has been mentioned
>before). Or maybe it means that df/dz and df/dz-bar exist, who knows. I
>doubt it has to do with R-differentiability, which you mentioned, which
>we haven't come across yet.
It's not entirely clear to me either. If you're familiar with the
notion, it probably means just "differentiable as a function of x
and y", meaning that for every point x+iy there is a real-linear
operator T:R^2 -> R^2 such that
f(x+s, y+t) = f(x,y) + T(s,t) + E(s,t),
where E is an error term satisfying E(s,t)/||(s,t)|| -> 0 as
(s,t) -> 0.
"R-differentiable" is not a standard term, it's just a word
I made up for that post to denote the concept defined above.
A standard term for it would be "Frechet differentiable", I
suppose, although it's usually just called "differentiable"
in advanced calculus.
(The reason I didn't want to just say "differentiable" is
that here "differentiable" could also mean "the limit
f'(z) = lim_{h->0} (f(z+h)-f(z))/h
exists" - that "differentiable" means something very different,
so I called it "C-differentiable" just to give it a name.)
>Anyway, thanks so far to both of you for your help.
>
>
>>>Incidentally, the definition of "holomorphic at a point" that
>>>I remember from school is that f is said to be holomorphic at
>>>a point if f is holomorphic on an open neighborhood of the point.
>
>This is the definition we've learned.
What book do you find this definition in? "Holomorphic at a
point" defined as above, _not_ "analytic at a point"?
???
************************
David C. Ullrich
Carlo Teubner
>>D then f is holomorphic on D." [D being an open disc]
>
>
> That's very different from your original
>
> "If at a point x + iy the Cauchy-Riemann equations hold,
> and u and v are continuous, then f is holomorphic at x + iy."
That was just an introductory statement (which I got wrong: it needs to
hold in an open subset, not a point), and not my actual question.
[snip helpful explanation of possible meaning of "differentiable"]
Ok, we haven't come across what you called R-differentiable; I'll leave
it for now and ask my tutor. Thanks.
>>>>Incidentally, the definition of "holomorphic at a point" that
>>>>I remember from school is that f is said to be holomorphic at
>>>>a point if f is holomorphic on an open neighborhood of the point.
>>
>>This is the definition we've learned.
>
>
> What book do you find this definition in? "Holomorphic at a
> point" defined as above, _not_ "analytic at a point"?
>
Hilary Priestly, _Introduction to Complex Analysis_, 2nd ed., p.59:
"A complex-valued function f is said to be holomorphic at a point a in C
if there exists r > 0 such that f is defined and holormorphic in D(a;r)."
Carlo
David C. Ullrich
signature."@example.com> wrote:
>>>"Show that if f is a differentiable function satisfying df/dz-bar = 0 on
>>>D then f is holomorphic on D." [D being an open disc]
>>
>>
>> That's very different from your original
>>
>> "If at a point x + iy the Cauchy-Riemann equations hold,
>> and u and v are continuous, then f is holomorphic at x + iy."
>
>That was just an introductory statement (which I got wrong: it needs to
>hold in an open subset, not a point), and not my actual question.
>
>[snip helpful explanation of possible meaning of "differentiable"]
>
>Ok, we haven't come across what you called R-differentiable; I'll leave
>it for now and ask my tutor. Thanks.
If you haven't got that notion then yes you should definitely
ask whoever assigned the problem exactly what the word
"differentiable" means in
"Show that if f is a differentiable function satisfying
df/dz-bar = 0 on D then f is holomorphic on D."
[D being an open disc]
Because as we've seen if it just means that the partial derivatives
exist then the exercise is wrong. It's right if differentiable
means what I've been calling R-differentiable. And it's right
if we assume that the partial derivatives of u and v are
continuous - that's the version you find in a lot of books.
But there's simply no way that "differentiable" _should_
mean that the partials are continuous - that's stronger
than differentiability.
>>>>>Incidentally, the definition of "holomorphic at a point" that
>>>>>I remember from school is that f is said to be holomorphic at
>>>>>a point if f is holomorphic on an open neighborhood of the point.
>>>
>>>This is the definition we've learned.
>>
>>
>> What book do you find this definition in? "Holomorphic at a
>> point" defined as above, _not_ "analytic at a point"?
>>
>
>Hilary Priestly, _Introduction to Complex Analysis_, 2nd ed., p.59:
>"A complex-valued function f is said to be holomorphic at a point a in C
>if there exists r > 0 such that f is defined and holormorphic in D(a;r)."
Huh. This seems simply wrong to me.
KRamsay
In article <bkf5t05073nqi3363...@4ax.com>, David C. Ullrich
<ull...@math.okstate.edu> writes:
|Good for Wikipedia!
Good for Axel Boldt, apparently!
|Assuming differentiability in this sense doesn't give the
|best results of the sort we're talking about here, but
|it _is_ the way to understand _pointwise_ differentiability.
Once I was doing some constructive analysis, and considering
the proof that a function whose derivative is 0 on a closed
interval is constant. It seemed much more awkward to assume
"pointwise" differentiability than something we could call
"uniform" differentiability, and a mathematician who does
constructive mathematics confirmed that one does indeed
generally prefer to use a definition of differentiability that is
"uniform differentiability on compact sets". And in more than
one independent variable, if we're going to be comparing
all sufficiently close pairs of points in the domain, the
comparison has to be by means of the "total" derivative.
The existence of partial derivatives comes off as a kind
of weird weakening.... I've wondered whether this
Looman-Menchoff theorem has been useful for later
results, or whether it's just satisfying an intellectual
curiousity about exactly what conditions are needed,
sort of like figuring out whether the commutative law
of addition is needed in the definition of "field".
|>Incidentally, the definition of "holomorphic at a point" that
|>I remember from school is that f is said to be holomorphic at
|>a point if f is holomorphic on an open neighborhood of the point.
|
|Are you sure you've seen this in a standard reference?
|I haven't - all I'm familiar with is that f is _analytic_
|at a point if it's holomorphic in a neighborhood of the
|point.
D'oh!
Is there much of a reason why these are separate terms?
I know people want to distinguish the differentiability definition
from the convergent power series definition, until they've been
shown equivalent, but is there more to it...?
Keith Ramsay
P.S. I also used to confuse the term "entire" with them!
KRamsay
signature."@example.com> writes:
|"Show that if f is a differentiable function satisfying df/dz-bar = 0 on
|D then f is holomorphic on D." [D being an open disc]
Well, it's clear that for the exercise to be correct, it must rule
out f(z)=e^{-1/z^4} if z<>0 and f(z)=0 if z=0 as a counterexample.
That's holomorphic on the plane minus the origin. At the origin
it has partial derivatives that are 0.
[...]
|One problem with the question is that it's not clear to me what
|"differentiable" means, though from context I might guess it means
|"differentiable as a function of x and y" (since that has been mentioned
|before).
Yes, I think so. And not just the existence of partial derivatives,
but of a "tangent plane approximation" at each point.
Keith Ramsay
David C. Ullrich
>
>In article <bkf5t05073nqi3363...@4ax.com>, David C. Ullrich
>[...]
>
>|>Incidentally, the definition of "holomorphic at a point" that
>|>I remember from school is that f is said to be holomorphic at
>|>a point if f is holomorphic on an open neighborhood of the point.
>|
>|Are you sure you've seen this in a standard reference?
>|I haven't - all I'm familiar with is that f is _analytic_
>|at a point if it's holomorphic in a neighborhood of the
>|point.
>
>D'oh!
You should note the question mark at the end of the
first sentence in the previous paragraph. I was surprised
that the OP came up with an actual reference. Hard to say
that that reference is _wrong_, but it's the only time
I've ever seen anyone define "holomoprhic" except in
the context of a function defined on an open set.
>Is there much of a reason why these are separate terms?
>I know people want to distinguish the differentiability definition
>from the convergent power series definition, until they've been
>shown equivalent, but is there more to it...?
My _guess_ is that it all started back in the days when
a "function" was necessarily given by a formula, or at
worst a few formulas on parts of the domain. Then an
analytic function was one that was equal to its power
series - distinctions like the difference between
"on E" and "in a neighborhood of E" would not have
come up, since a function didn't necessarily even
have a domain, other than the sense in which a
calculus student uses the term at present, the
set where the formula makes sense.
Then later when modern terminology about functions
arises people realize that what they really meant
by "analytic on E" was "is a power series near
every point of E", which turns out to be equivalent
(in the plane) to "is differentiable in a neighborhood
of E". And then people notice that this notion of
"analytic" kind of sucks - the definition is needlessly
complicated, and for example E and F disjoint, f analytic
on E and analytic on F does not imply f analytic on
E union F. Someone decides that "holomorphic in O"
(for open sets O) serves the same purpose with
fewer complications.
Just a guess.
>Keith Ramsay
>
>P.S. I also used to confuse the term "entire" with them!
************************
David C. Ullrich
KRamsay
<ull...@math.okstate.edu> writes:
|My _guess_ is that it all started back in the days when
|a "function" was necessarily given by a formula,
|Then later when modern terminology about functions
|arises people realize that what they really meant
|by "analytic on E" was "is a power series near
|every point of E", which turns out to be equivalent
|(in the plane) to "is differentiable in a neighborhood
|of E". And then people notice that this notion of
|"analytic" kind of sucks - the definition is needlessly
|complicated, and for example E and F disjoint, f analytic
|on E and analytic on F does not imply f analytic on
|E union F. Someone decides that "holomorphic in O"
|(for open sets O) serves the same purpose with
|fewer complications.
I'm afraid I don't follow this remark about f analytic on
E and F not implying f analytic on E union F. Surely
f can't be a power series near each point of E and a
power series near each point of F without being a power
series near each point of the union of E and F. I don't
see what else would make sense here. The restriction
of f to E and the restriction of f to F being separately
analytic, in this sense, doesn't mean that the restriction
of f to the union of E and F is analytic, but it also doesn't
imply even if f is defined on the whole plane that f is
differentiable on a neighborhood of E or F, e.g. f(z)=Re(z)
restricted to straight lines.
Keith Ramsay
David C. Ullrich
>In article <dssft01rr5c4fvt6m...@4ax.com>, David C. Ullrich
><ull...@math.okstate.edu> writes:
>|My _guess_ is that it all started back in the days when
>|a "function" was necessarily given by a formula,
>[...]
>|Then later when modern terminology about functions
>|arises people realize that what they really meant
>|by "analytic on E" was "is a power series near
>|every point of E", which turns out to be equivalent
>|(in the plane) to "is differentiable in a neighborhood
>|of E". And then people notice that this notion of
>|"analytic" kind of sucks - the definition is needlessly
>|complicated, and for example E and F disjoint, f analytic
>|on E and analytic on F does not imply f analytic on
>|E union F. Someone decides that "holomorphic in O"
>|(for open sets O) serves the same purpose with
>|fewer complications.
>
>I'm afraid I don't follow this remark about f analytic on
>E and F not implying f analytic on E union F.
Let E = [0,1), F = [1,2). Say f is identically 0 on
E and f is identically 1 on F. Then f is analytic on
E (or probably I should say the restriction of f to
E is analytic on E, which might have made things
more clear), f is analytic on F but f is not
analytic on the union.
>Surely
>f can't be a power series near each point of E and a
>power series near each point of F
But "f is a power series near each point of E" is
not quite the definition - f need not even be
_defined_ near points of E. The definition is
that for every x in E there is a power series
centered at x which equals f at points of E
near x.
Hmm, come to look at it, "f is a power series near each point of E"
is exactly what I wrote. That was a slightly informal version...
>without being a power
>series near each point of the union of E and F. I don't
>see what else would make sense here. The restriction
>of f to E and the restriction of f to F being separately
>analytic, in this sense, doesn't mean that the restriction
>of f to the union of E and F is analytic, but it also doesn't
>imply even if f is defined on the whole plane that f is
>differentiable on a neighborhood of E or F, e.g. f(z)=Re(z)
>restricted to straight lines.
>
>Keith Ramsay
************************
David C. Ullrich