Functions Definition and 1000 Threads

For example, I am following the below proof: Although the above derivation involves a projection on the position basis, it appears one can generalize this result by using any complete basis. So despite it not being explicitly mentioned here, are all wave functions with any continuum basis...

Is there a difference between the linear independence of ##\{x,e^x\}## and ##\{ex,e^x\}##? It can be shown that both only have the trivial solution when represented as a linear combination equal to zero. However, the definition of linear independence is: "Two functions are linearly independent...

Homework Statement I need to draw this function: however I don't get how? I have the solution but I don't understand how do I get that from the given function. Someone please try to explain? Thanks

Homework Statement I'm searching for the integral that gives arcosu Homework Equations as we know : ∫u'/[1-u^2]^0.5 dx = arcsinu derivative of arccosu = -u'/[1-u^2]^0.5 + C derivative of arcsinu= u'/[1-u^2]^0.5 The Attempt at a Solution when I type the -u'/[1-u^2]^0.5 on the online integral...

Hey! :o Let $\text{Val} = \{0, 1\}^8$, $\text{Adr} = \{0, 1\}^{32}$ and $\text{Mem} = \text{Val}^{\text{Adr}}$. The addition modulo $2^8$ of two numbers in binary system of length $8$, is given by the mapping: $$\text{add}_{\text{Val}} : \text{Val}\times \text{Val}\rightarrow \text{Val} \\...

Hi folks, When you have a differential equation and the unknown function is complex, like in the Schrodinger equation, What methods should you use to solve it? I mean, there is a theory of complex functions, Laurent series, Cauchy integrals and so on, I guess if it would be possible to...

Homework Statement Can someone check my work? Homework EquationsThe Attempt at a Solution 1. ##\frac{1}{s+2}+\frac{1}{s^2+1}## 2. ##\frac{2}{s}+\frac{3}{s+4}## 3. ##\frac{s*sin(-2)+cos(-2)}{s^2+1}## 4. ##\frac{1}{(s+1)^2}## 5. Don't really know how to do this one...

Hey! :o Could you give me an example of a strong concave function $f:[0,3]\rightarrow \mathbb{R}$ that is not continuous? (Wondering) We have that $f''(x)<0$. Since the function has not to be continuous, the derivatives are neither continuous, are they? (Wondering) Is maybe the...

Hello! (Wave) We consider the following Cauchy problem $u_t=u_{xx} \text{ in } (0,T) \times \mathbb{R} \\ u(0,x)=\phi(x) \text{ where } \phi(x)=-\phi(-x), x \in \mathbb{R} $ I want to show that $ u(t,0)=0, \forall t \geq 0 $. We have the following theorem: Let $\phi \in...

Green functions are defined in mathematics as solutions of inhomogeneous differential equations with a dirac delta as the right hand side and are used for solving such equations with a generic right hand side. But in QFT, n-point correlation functions are also called Green functions. Why is...

I thought I understood the concept of a correlation function, but I having some doubts. What exactly does a correlation function quantify and furthermore, what is a correlation length. As far as I understand, a correlation between two variables ##X## and ##Y## quantifies how much the two...

If ##t## is a function of ##r##, then we may in theory find ##r## as a function of ##t##, as claimed in the last paragraph of the attachment below. My issue is this is only true if ##t## is a 1-1 function of ##r##. Otherwise, suppose ##t=r^2##. Then ##r=\pm\sqrt{t}##, which isn't a function. I...

In my calculus textbook, it shows that a function's solution can be approximated using an approximated function tangent to the original function based on an approximated solution, where the equation to find the approximated is L(x) = f(X0) + f'(X0)*(X-X0), where when rearranged, gives x = Xo -...

Homework Statement Evaluate the following line integrals in the complex plane by direct integration. Homework Equations Z= x+i y = Cos(θ) +i Sin(θ) = e^i*θ The Attempt at a Solution I'm not sure how to evaluated this by hand. I tried using Z= x+i y = Cos(θ) +i Sin(θ), and evaluating the...

Homework Statement Derive the transfer function for both circuits \frac{V_{out}}{V_{in}} sketch Bode plots for each circuit (amplitude and phase) Homework Equations Z_c=\frac{1}{j{\omega}C}~and~{\omega}_C=\frac{1}{RC} The Attempt at a Solution We can treat this as a potential divider using the...

Are there differential equations that, for some reason, don't have a Green function? Are there conditions for a DE to satisfy so that it can have a Green function? Thanks

In functional analysis functions ##f## and ##g## are orthogonal on the interval ##[a,b]## if \int^b_a f(x)g(x)dx=0 But what if we have functions of two variables ##f(x,y)## and ##g(x,y)## that are orthogonal on the interval ##[a,b]##. Is there some definitions \int^b_a f(x,z)g(z,y)dz=0?

Hello! (Wave) I have a question about the proof of the maximum principle for subharmonic functions. The maximum principle is the following: The subharmonic in $\Omega$ function $v$ does not achieve its maximum at the inner points of $\Omega$ if it is not constant. Proof: We suppose that at...

sin\theta 3/5

Suppose u(x,y) and v(x,y) are harmonic on G and c is an element of R. Prove u(x,y) + cv(x,y) is also harmonic. I tried using the Laplace Equation of Uxx+Uyy=0 I have: du/dx=Ux d^2u/dx^2=Uxx du/dy=Uy d^2u/dy^2=Uyy dv/dx=cVx d^2v/dx^2=cVxx dv/dy=cVy d^2v/dy^2=cVyy I'm not really sure how to...

Mentor note: moved to homework section y = sin(x) y = cos(x) y = tan(x) y = csc(x) y = sec(x) y = cot(x) (a) 0 (b) 4 (c) 6 (d) 2 I thought it was (c) because i graphed all the trig functions and they passed the vertical line test but the answer sheet is saying (d) 2

Homework Statement If ##r_1, r_2, r_3## are distinct real numbers, show that ##e^{r_1t}, e^{r_2t}, e^{r_3t}## are linearly independent. Homework EquationsThe Attempt at a Solution By book starts off by assuming that the functions are linearly dependent, towards contradiction. So ##c_1e^{r_1t}...

Given the theory $$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}m_{\phi}^{2}\phi^{2}+\partial_{\mu}\chi^{*}\partial^{\mu}\chi-m_{\chi}^{2}\chi^{*}\chi+\mathcal{L}_{\text{int}},\qquad \mathcal{L}_{\text{int}}=-g\phi\chi^{*}\chi,$$ the time-correlation function ##\langle \Omega |...

Hi all, I understand that the mixed partial derivative at some point may not be equal if the such mixed partial derivative is not continuous at the point, but are the actual functions of mixed partial derivatives always equal? In other words, if I simply compute the mixed partial derivatives...

Hello all, First of all, I am aware that dissonance and consonance between pitches also depend to an extent by culture and musical origin but there also seems to be some degree of objective perception among people that can be explained scientifically. Also, I'm very new to this so I could be...

Homework Statement The water depth in a harbor is 21m at high tide and 11m at low tide. Once cycle is completed every 12 hrs. (a) Find equation for the depth as a function of time. (b) Draw a graph for 48 hrs after low tide, which occurred at 14:00. (c) State the times where the water...

Homework Statement Find the following limit: Homework EquationsThe Attempt at a Solution My lecturer has said that rational functions which are a ratio of two polynomials are continuous on R^2. He also said that the limits of continuous functions can be computed by direct substitution. The...

first of all assume that I don't have proper math knowledge. I came across this idea while I was studying last night so I need to verify if it's valid, true, have sense etc. orthogonality of function is defined like this: https://en.wikipedia.org/wiki/Orthogonal_functions I wanted to...

Hi, As is commonly known, u = u(T,v) h = u(T,p) I've worked with some maths proofs of this a while ago, but do you guys have an intuitive way of understanding this without the maths, that is, why the state function for internal energy is defined by intensive volume and enthalpy with pressure...

What does it mean for a ##f(x,y)## to be differentiable at ##(a,b)##? Do I have to somehow show ##f(x,y)-f(a,b)-\nabla f(a,b)\cdot \left( x-a,y-b \right) =0 ##? To show the function is not though, it's enough to show, using the limit definition, that the partial derivative approaching in one...

I've been reading about canonical transformations in Hamiltonian mechanics and I'm a bit confused about the following: The author considers a canonical transformation $$q\quad\rightarrow\quad Q\quad ,\quad p\quad\rightarrow\quad P$$ generated by some function ##G##. He then considers the case...

The questions is asking me to find \frac{f}{g} basically , the question is asking me to find the answer , even though i know it, i can't get my head around it. the composite function is f(x)=x^2+1 g(x)=1/x we need to find foG (f of g) [composite functions].

Homework Statement Let ##f(t)=\int_{t}^{t^2} \frac{1}{s+\sin{s}}ds,t>1.##Express ##f## as a composition of two differentiable functions ##g:ℝ→ℝ^2## and ##h:ℝ^2→ℝ##. In addition, find the derivative of ##f## (using the composition). Homework EquationsThe Attempt at a Solution Honestly, I have...

Hi I am writing my final Mathematics exams for Grade 12 in South Africa in 5 days. I am well prepared with an aim of getting 100%, but one concept in functions might prevent that - the concept of how the nature of roots are affected by vertical/horizontal shifts in a function, and how to...

Temperatures can be converted from Fahrenheit to Celsius using the function f(x) = 5 /9 (x − 32). (a) Calculate f(59). (b) Find f −1 (x), and verify that f −1 (f(59)) = 59. (c) Let K be the set {x : f(x) = x}. Find all elements of K and list K

Homework Statement I have a series of 12 values that I need to calculate the Theoretical Intensity, I, using the formula below. I have found values for all variables and their uncertainties, and have calculated the I value for each set using the formula. Now I need to calculate the...

Hi PF! I am wondering why we define velocity for polar coordinates as $$\vec{V} = \nabla \times \frac{\psi(r,z)}{r} \vec{e_\theta}$$ and why we define velocity in spherical coordinates as $$\vec{V} = \nabla \times \frac{\psi(r,\phi)}{r \sin \phi} \vec{e_\theta}$$ The only thing I don't...

Homework Statement Show that the tangent to ##x^2-y^2=1## at points ##x_1=\cosh (u)## and ##y_1=\sinh(u)## cuts the x-axis at ##{\rm sech(u)}## and the y-axis at ##{\rm -csch(u)}##. Homework Equations Hyperbolic sine: ##\sinh (u)=\frac{1}{2}(e^u-e^{-u})## Hyperbolic...

Prove: ##(\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)## Newton's binomial: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n## and: ##(a-b)^n~\rightarrow~(-1)^kC^k_n## I ignore the coefficients. $$(\cosh(x)+\sinh(x))^n=\cosh^n(x)+\cosh^{n-1}\sinh(x)+...+\sinh^n(x)$$...

Here is the question: This is the step I came to after taking the derivatives and doing some simplification: ^ I did the work myself on paper, I just couldn't type out the whole thing clearly so that anyone else can see what I'm referring too... so I used some online tool to show that...

Homework Statement Design an combinational circuit using a decoder and external gates defined by the boolean functions F1, F2, F3(see picture) Homework EquationsThe Attempt at a Solution I'm quite confused as to the exact method in doing this. I understand that a decoder takes n inputs and...

Hi! For the probability interpretation of wave functions to work, the latter have to be square integrable and therefore, they vanish at infinity. I'm reading Gasiorowicz's Quantum Physics and, as you can see in the attached image of the page, he works his way to find the momentum operator. My...

I have this exercise on my book and I believe it is quite simple to solve, but I'm not sure if I did good, so here it is Homework Statement given a vector B ∈ ℝn, B ≠ 0 and a function F : ℝ → ℝn such that F(t) ⋅ B = t ∀t and the angle φ between F'(t) and B is constant with respect to t, show...

On wikipedia it says the following, "...the Green's function of the Hamiltonian is a key concept with important links to the concept of density of states." https://en.wikipedia.org/wiki/Green%27s_function Can anyone explain why?

Homework Statement I have the two functions below and have to find the convolution \beta * L Homework Equations Assume a<1 \beta(x)=\begin{cases} \frac{\pi}{4a}\cos\left(\frac{\pi x}{2a}\right) & \left|x\right|<a\\ 0 & \left|x\right|\geq a \end{cases} L(x)=\begin{cases} 1 &...

When working in the complex domain (##z = x + iy##), how does one write the equation of a line? I have attached a problem I was working on (and have the solution), but am curious as to why the definition of a line is given by ##ax + by = c##. Are not ##x## and ##y## also variables that take on...

$\textbf{10)} \\ f(x)\text{ is continuous at all } \textit{x} \\ \displaystyle f(0)=2, \, f'(0)=-3,\, f''(0)=0 $ $\text{let} \textbf{ g } \text{be a function whose derivative is given by}\\ \displaystyle g'(x)=e^{-2 x} (3f(x))+2f'(x) \text{ for all x}\\$ $\text{a) write an equation of the...

Is v(r) ≡ v(x,y,z)

I have some conceptual issues with functions in vectors spaces. I don't really get what are really the components of the vector / function. When we look at the inner product, it's very similar to dot product, as if each value of a function was a component : So I tend to think to f(t) as the...

Through experimental observations, I have found that the two functions ##f\left(x\right)=x^{a\left(a-x^3\right)}-a## and ##g\left(x\right)=a^{x\left(x-a^3\right)}-x## will always intersect at ##a## when ##x>0##. Is there a way to mathematically prove this? For instance, simultaneously solving...