Homework Statement
State, with justification, if the Fundamental Theorem of Contour Integration can be applied to the following integrals. Evaluate both integrals.
\begin{eqnarray*}
(i) \hspace{0.2cm} \int_\gamma \frac{1}{z} dz \\
(ii) \hspace{0.2cm} \int_\gamma \overline{z} dz \\...
Ah, because an open interval can be written as the countably infinite intersection of closed intervals...
Hmm, ok, let me think about it some more :) Thanks for your help!
Hi Krylov/Andrew,
If I take the (finite) union of n number of intervals on the real line in the form (-∞,a], then the resultant union is going to be in the form (-∞,a], yes? Because there will always be some a' that is bigger than all the other a's, hence it will be the right-most interval for...
Thanks for your response, Andrew!
a counterexample - in this case a countable union that is not in the set: To get this, as the outcome we would have to have some set that was in the form (-∞,a) (i.e. right interval is open) but that would mean there is some set in our union of this form, which...
Homework Statement
Let ε = { (-∞,a] : a∈ℝ } be the collection of all intervals of the form (-∞,a] = {x∈ℝ : x≤a} for some a∈ℝ.
Is ε closed under countable unions?
Homework Equations
Potentially De Morgan's laws?
The Attempt at a Solution
Hi everyone,
Thanks in advance for looking at my...
I thought it might come out looking like an 'In'... :)
So if I get it down to:
p^{a}(1-p)^{b} = e^c
how can I solve for p then? Assuming the a and b are large-ish numbers like, say, 10 and 20.
Homework Statement
(Here, by 'log' I mean natural logarithm)
Solve for x:
a.log(x) + b.log(1-x) = c
for a, b and c constants
Homework Equations
The Attempt at a Solution
Hi everyone,
This is so embarrassing but this is really stumping me! I know how to do it if a=b...
Actually, I have had a wave of inspiration since - is this correct?
The singularities occur for 2.cos(z)-1 = 0 i.e. cos(z) = 1/2
This happens for z = pi/3 (+ 2k.pi, but this z is the smallest one)
So then
the distance from z=1 to z=pi/3 is:
√(1 - pi/3)^2) = 2pi/3
which is then...
Homework Statement
Find the radius of convergence of the Taylor series at 0 of this function
f(z) = \frac{e^{z}}{2cosz-1}
Homework Equations
The Attempt at a Solution
Hi everyone,
Here's what I've done so far:
First, I tried to re-write it as a Laurent series to find...
Thanks for your reply!
Ah. So the singularity would be at z=0 then?
So the distance from z=0 to z=1 would be 1, so the radius of convergence is then R=1 centered at the point z=1?
Homework Statement
Find the radius of convergence of the Taylor series at z = 1 of the function:
\frac{1}{e^{z}-1}
Homework Equations
The Attempt at a Solution
Hi everyone,
Here's what I've done so far.
Multiply top and bottom by minus 1 to get:
-1/(1-e^z)
And then...
Homework Statement
Hi everyone,
I'm just wondering if someone could please clarify for me what infinity to the power of zero is? I seem to be finding conflicting opinions about this online. Is it '1' or 'not defined'?
Thanks!
Homework Equations
The Attempt at a Solution
Homework Statement
Hi everyone,
This is more of a definition clarification than a question. I'm just wondering if a branch is the same thing as a branch line/branch cut? I've come across a question set that is asking me to find branches, but I can only find stuff on branch lines/cuts and...
Hi,
Thanks for your reply! So then I would have:
Assume λ<0
Let λ = -a^2, for some real, non-zero number a
Then
y'' + (a^2)y = 0
y(t) = c1.sin(at) + c2.cos(at)
y(0)=0, therefore c2 = 0
y(1)=0, therefore c1 = 0 (trivial solution)
or sin(a) = 0
a = n∏
so then λ =...
Homework Statement
Solve:
y'' - λy = 0
where y(0)=y(1)=0, y=y(t)
Homework Equations
The Attempt at a Solution
Hi everyone,
This is part of a PDE question, I just need to solve this particular ODE. I know how to do it in the case for y'' + λy = 0, where you get the...
Homework Statement
Let λ_n denote the nth eigenvalue for the problem:
-Δu = λu in A, u=0 on ∂A (*)
which is obtained by minimizing the Rayleigh quotient over all non-zero functions that vanish on ∂A and are orthogonal to the first n-1 eigenfunctions.
(i) Show that (*) has no...
Homework Statement
Hi, this is part of a stats problem, in the solutions they go from:
d/dc of ln(1-0.5^c)
then next line they have:
ln(0.5).[0.5^c]/(1-0.5^c)
I don't understand how they did this!
Homework Equations
The Attempt at a Solution
So, I know that the...
Homework Statement
The logistic random variable X has CDF:
F(X) = exp(a + bx)/(a + exp(a + bx)), b>0
(i) Obtain a formula for random values of X in terms of R ~ U(0,1)
(ii) Using the cdf, suppose that a certain application requires you to generate 'the middle one of three' i.i.d...
Homework Statement
Consider the following probability table:
X 1 2 3 4
P(X) 0.4 0.25 0.25 0.1
Use the rejection method to generate a random number.
Use the following list of random numbers:
0.6072, 0.4893, 0.0899, 0.3456, 0.4419...
Homework Statement
Hi everyone,
I'm working through some max likelihood questions and am badly stuck on this one. Please could you take a look at what I'm doing and tell me if I'm going in the right direction?
Q. Team A and B play two games of soccer, each game having two halves of equal...
Homework Statement
Hi everyone, this is part of a longer question, but the current part involves me finding the value of c in the following equation:
L = c^3.(0.5)^(c-1).(0.6)^(c-1).(0.8)^(c-1).(1- (0.5)^c)^42.(0.5)^(8c)
Homework Equations
The Attempt at a Solution
Here's what...
Homework Statement
A bag contains sequentially numbered lots (1,2...N). Lots are drawn at random (each lot has the same probability of being drawn). Two lots are drawn without replacement and are observed to be X_1 = 17 and X_2 = 30. What is the MLE of N, the number of lots in a bag...
Homework Statement
Lifetimes of components are Gamma distributed. The parameters of the Gamma are
shape = a
scale = λ
The pdf is:
f(x) = (λ^a).x^(a-1).e^(-λx)/Γ(a)
In this case, it is known that a = 3. Obtain the MLE of λ.
Homework Equations
The Attempt at a Solution
Hi...
Homework Statement
Let X be a first countable space where no sequence has more than one limit. Show that X must be Hausdorff.
Homework Equations
The Attempt at a Solution
Hi everyone,
Here's what I've done so far:
I used this thm: If X is a Hausdorff space, then sequences in...
Think again about how you are setting up the triangle. Look at it from a 'side-on' view i.e. you are seeing the side of the man's face, as opposed to viewing it from the front.
Which piece represents the hypotenuse? Is it the height of the man's hand from the water, the rope, or the water...
Homework Statement
Let (X,Ʈ) be a topological space and T \subseteq X a compact subset.
Show that T is compact as a subset of the space (T,Ʈ_T) where Ʈ_T is the relative topology on T.
Homework Equations
The Attempt at a Solution
Hi everyone,
Here's what I've done so far:
T...
Homework Statement
In the metric space (R^2, d), where d is the standard euclidean metric, show that the square {(x,y) : |x|< 1, |y|< 1} is open.
Homework Equations
A set is closed if and only if its complement is open.
A set is open if every point in the set is an interior point...
Homework Statement
Suppose that f(z) = ∑a_j.z^j for all complex z, the sum goes from j=0 to infinity.
(a) Find the power series expansion for f'
(b) Where does it converge?
(c) Find the power series expansion for f^2
(d) Where does it converge?
(e) Suppose that f'(x)^2 + f(x)^2 = 1...
Homework Statement
Q. (a) State Liouville's Theorem
(b) Suppose that f is analytic in C and satisfies f(z + m + in) = f(z) for all integers m,n . Prove f is constant.
Homework Equations
The Attempt at a Solution
(a) Liouville's Theorem - If f is bounded and analytic in C, then...
Homework Statement
Hi, I know this is a mechanics question, but I don't think the actual problem I have with it involves any mechanics, it's just integration techniques.
Find the deflection angle of a particle moving in the following repulsive central field:
U = α/r², α > 0...
Homework Statement
Find the deflection angle of a particle moving in a repulsive Coulomb field.
Homework Equations
The Attempt at a Solution
Hi everyone,
I can do this no problem if I just knew the equation for a repulsive Coulomb field! Can anyone please help me with this?
Thanks
Homework Statement
A function F of n real variables is called homogeneous of degree r if it satisfies
F(tx_1, tx_2, ..., tx_n) = (t^r)F(x_1,x_2,...,x_n)
By differentiation with respect to t, show that a function F is an eigenfunction of the operator:
x^1 ∂/∂x_1 + ... + x^n ∂/∂x_n...
Homework Statement
Suppose X = [0,1] x [0,1] and d is the metric on X induced from the Euclidean metric on R^2. Suppose also that Y = R^2 and d' is the Euclidean metric. Is the mapping
T: [0,1] x [0,1] \rightarrow R^2, T(x,y) = (xy, e^(x.y))
uniformly continuous? Explain your answer...
Homework Statement
Give an example of two norms on a vector space that are not equivalent.
Homework Equations
The Attempt at a Solution
Hi everyone,
I know the definition for equivalent norms. I also know that norms on a finite dimensional vector space are equivalent. So...
Homework Statement
For sets E,F \in L, show that χ_E = χ_F almost everywhere if and only if µ(EΔF) = 0
where χ_E is the characteristic function w.r.t. E
and µ(EΔF) is the lebesgue measure of the symmetric difference of E and F
and L is the set of lebesgue measurable sets
Homework...
Homework Statement
A point in a mechanics problem where I have to solve the ODE (dS/dx)^2 + mw^2x^2 = a
where m,w^2 are constants
Homework Equations
The Attempt at a Solution
Hi everyone,
We haven't actually covered how to solve these in my ODEs class yet (obviously my...
Homework Statement
Consider the vector field:
F = r/r3
where r = xi + yj + zk
Compute the flux of F out of a sphere of radius a centred at the origin.
Homework Equations
The Attempt at a Solution
Hi everyone,
I have: flux = \intF.dA
I can't use Gauss' Law, because the...
Homework Statement
Is the following ODE linear? If so, is it homogeneous?
xy'' + siny = 0, where y = y(x)
Homework Equations
Linear = coefficients of unknown function y(x) and its derivatives only depend on x, not the unknown
Homogeneous: can be written in form y'' + p(x)y' + q(x)y...
Homework Statement
Find all solutions of the equation:
y' = (2y)/(t.logt) = 1/t, t > 0
Homework Equations
Integrating factor I = exp(\intp(x)dx)
where y' + p(x)y = q(x)
The Attempt at a Solution
Hi everyone, here's what I've done so far:
Let p(t) = -2/(t.logt)
I =...
Homework Statement
Question 1 on the following page: http://www.maths.tcd.ie/~frolovs/Mechanics/PS10.pdf
It's the second part I'm stuck on ('Explain why the equations of motion do not...')
Homework Equations
The Attempt at a Solution
I first found equations for x_1 'dot' and...
Homework Statement
Homework Equations
The Attempt at a Solution
Hi, what is the correct definition for a Poisson bracket?
Some books say it is:
{f,g} = df/dp.dg/dq - df/dq.dg/dp
but others say it is:
{f,g} = df/dq.dg/dp - df/dp.dg/dq
One is the other multiplied...
Choose the subject that you like so much that you read about it in your spare time. Just liking to study a subject, but having no real interest in the subject itself, is not really enough if you want to be happy in your degree in college, as 3rd level is quite different to 2nd level. I speak...