Homework Statement
Describe how to determine whether an equilibrium is stable or unstable when [d2U/dx2]_0 = 0
From Classical Dynamics - Ch 2 #45 - Marion Thornton
2. Homework Equations AND 3. The Attempt at a Solution
When second derivative positive, equilibrium is stable. When...
I am actually looking for a proof as well.
You say that is the definition of the Taylor series, but how does one prove that if a function F is analytic, it can be represented by a power series of the form
\Sigma^{\infty}_{n=0}a_nz^n
where
a_n = f^{(n)}(0)/n!
My teacher recommended a...
I am pretty sure I'm in the same class as this person...except the no-textbook thing threw me off. (My class has a textbook, albeit one the professor doesn't seem to use.) If we're not in the same class, I am having trouble with the exact same problem, which I didn't think came from a book...I...
I'm not sure what you're saying here. I think you get that they don't exist or are undefined, because there would be 0 on the bottom of a fraction? But I'm not sure where you're going with this.
Just to go ahead and try this: would partial u partial x be y(xy)^(-1/2)?
And then partial u partial y be x(xy)^(-1/2)?
And both partials of v be 0? This is assuming that sqrt(xy) is just the 'real' part...if f(z) takes the form u + iv.
I have a feeling this is wrong, since Cauchy-Riemann is...
I mixed up my original message. I actually got x for the imaginary part and x_0 + 2x + iy for the real part. I still don't see how the x_0 was eliminated in your version of the expansion, above.
But the main issue is the fact that I get different values in the second part, evaluating the...
Wouldn't that mean the Cauchy-Riemann equations don't hold? I'm a little unsure on what u would be in this case.
Do I need to separate sqrt(|xy|) into the real and imaginary parts? Or can I just assume all is real and then take the partial derivates of u, and the partials of v would just be...
Homework Statement
In the title: f(z) = sqrt(|xy|)...show that this satisfies the Cauchy-Riemann equations at z=0, but is not differentiable there.
Homework Equations
Cauchy-Riemann just states that partial u partial x = partial v partial y and partial u partial y = - partial v partial...
Homework Statement
Show that f(z) = zRez is differentiable only at z=0,
find f'(0)
The Attempt at a Solution
This should be easy. I find the limit as z_0 approaches 0 of [f(z+z_0) - f(z)]/(z_0) for this function...expand it out, simplify, and find what the limit is when z_0 is...
This isn't a homework question; I'm sorry if I'm mis-posting, but I thought someone here could help.
See this link:
http://books.google.com/books?id=1QxenjJL6i0C&printsec=frontcover&dq=intro+complex+analysis&lr=&as_brr=3#PPA56,M1
On page 56, does this book have a misprint in the...
Homework Statement
The functions Re(z)/|z|, z/|z|, Re(z^2)/|z|^2, and zRe(z)/|z| are all defined for z!=0 (z is not equal to 0)
Which of them can be defined at the point z=0 in such a way that the extended functions are continuous at z=0?
It gives the answer to be:
Only f(z)=zRe(z)/|z|...
So b would be 1/m, i/n, and 0?
And c would be 1+i (all go to infinity), 1, 0, p/m, and iq/n?
What about a? Are there no limit points? It doesn't seem to converge anywhere. Except maybe at 1 and 2, when n goes to infinity?
Homework Statement
Find the limit points of the set of all points z such that:
a.) z=1+(-1)^{n}\frac{n}{n+1} (n=1, 2, ...)
b.) z=\frac{1}{m}+\frac{i}{n} (m, n=+/-1, +/-2, ...)
c.) z=\frac{p}{m}+i\frac{q}{n} (m, n, p, q=+/1, +/-2 ...)
d.) |z|<1
Homework Equations
None.
The Attempt at a...
I see your point, but I don't know how to generalize the value for the dimension.
From what you're saying: in part a.), the basis is simply the set of {1, x, x^2...x^(N-1)} and teh vectors are represented by the coefficients {c_0, c_1...c_(N-1)}. Okay, that makes sense.
For part b.), the...
Thanks for the tip.
I must be confused from what my teacher's notes are saying. He basically said the vectors would be defined as polynomials, like your p(x) above.
And this sentence:
"Can you suggest a set of simple functions of x that you can combine with constant coefficients that are...