In mathematics, a function is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. It is customarily denoted by letters such as f, g and h.If the function is called f, this relation is denoted by y = f (x) (which reads "f of x"), where the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (e.g., f is a function of the variable x).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means of illustrating the function.
Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.
I operated by placing ##S## and ##T## to two singlets belonging to ##X## and therefore established that for ##T, S \in X##, therefore ##f (T) = f (S) \implies S = T##, consequentially: $$f (T \cap S ) = f (T \cap T) = f (T) \cap f (T) = f (T) \cap f (S)$$. I would like to know if my procedure is...
In the following I ask WA to solve the given equation and it produces a solution using the Lambert W function.
I thought : $$W(x*e^x) = x$$ but here it seems $$W_n \left(\frac{-MT}{P}*e^{\frac{-MT}{P}}\right) \neq \frac{-MT}{P}$$
Is there a difference between ##W(x)## and ##W_n(x)## ?
Is there a function that outputs a 1 when the input is a multiple of a number of your choice and 0 if otherwise. The input is also restricted to natural numbers.
The only thing I can come up with is something of the form:
f(x) = [sin(ax)+1]/2
but this does not output a 0 when I want it.
Let ##0 < \alpha < 1##. Find a necessary and sufficient condition for the function ##f : [0,1] \to \mathbb{R}##, ##f(x) = \sqrt{x}##, to belong to the class ##C^{0,\alpha}([0,1])##.
Hi there!
I would like to know if the following simplification is correct or not:
Let A be a function of x, y, and z
$$\frac{\partial^2A}{\partial x^2}+\frac{\partial^2A}{\partial y^2}$$
$$=\ \frac{\partial^2A\partial y^2+\partial^2A\partial x^2}{\partial x^2\partial y^2}$$...
Hello everybody,
I have a question regarding this visualization of a multidimensional function. Given f(u, v) = e^{−cu} sin(u) sin(v). Im confused why the maximas/minimas have half positive Trace and half negative Trace. I thought because its maxima it only has to be negative. 3D vis
2D...
Let's see how messy it gets...
##\dfrac{dy}{dx}=\dfrac{(1-10x)(\sqrt{x^2+2})5x^4 -(x^5)(-10)(\sqrt{x^2+2})-x^5(1-10x)\frac{1}{2}(x^2+2)^{-\frac{1}{2}}2x}{[(1-10x)(\sqrt{x^2+2})]^2}##...
As a follow up for : https://www.physicsforums.com/threads/let-k-n-show-that-there-is-i-n-s-t-1-1-k-i-1-2-k-i-1-4.1054669/
show that ## \alpha\left(k\right)\ :=\ \left(1-\tfrac{1}{k}\right)^{\ln\left(2\right)k}-\left(1-\tfrac{2}{k}\right)^{\ln\left(2\right)k} ## is decreasing for ##...
Given a function F(x,y)=A*x*x*y, calculate dF(x,y)/d(1/x), to calculate this derivative I make a change of variable, let u=1/x, then the function becomes F(u,y)=A*(1/u*u)*y, calculating the derivative with respect to u, we have dF/du=-2*A*y*(1/(u*u *u)) replacing we have dF/d(1/x)=-2*A*x*x*x*y...
I ran across the following problem :
Statement:
Consider a gas of ## N ## fermions and suppose that each energy level ## \varepsilon_n## has a multiplicity of ## g_n = (n+1)^2 ##. What is the Fermi energy and the average energy of this gas when ## N \rightarrow \infty## ?
My attempt:
The...
At first I thought that this force vector ## \vec F = 3 \hat x + 2 \hat y ## is a function of ## x ## and ## y ##, which is to say that its magnitude and direction vary with the x and y positions, but this is not so, right? It's just a force with a constant magnitude and direction.
And I can...
I am aware that this question is very simple and basic.
Using ##y(t)=y_0+v_{0,y}t-\frac {1}{2}gt^2## we can find distance as a function of time:
##|y_1-y_2|=|y_0+v_{0,y}t|=-y_0- v_{0,y}t##
I assumed the downward direction to be negative. So as I wrote ##D(t)=-y_0- v_{0,y}t##. It tells that the...
My main question here is about how we actually justify, hopefully fairly rigorously, the steps leading towards converting the sum to an integral.
My work is below:
If we consider the canonical ensemble then, after tracing over the corresponding exponential we get:
$$Z = \sum_{n=0}^\infty...
Please walk me step by step on how to do it (we don't have imaginary numbers so don't bring that up)
Also how to put signs on the numbers line when I get minus in the root? (non solveable equation)
Sorry for my English.
For this,
I first try to work out where function is increasing
My working is
##f'(x) = 12x^3 - 12x^2 - 24x##
For increasing,
##12x(x^2 - x - 2) > 0##
##12x > 0## and ##(x - 2)(x + 1) > 0##
##x > 0## and ##x > 2## and ##x > -1##
However, how do I combine those facts into a single domain...
I am trying to write a python script to plot the function,
Where
##V_0 = 5~V##
##t_0 = 10~ms##
##\tau = 5~ms##
My script that I have written to try to do this is,
Which plots,
However, the plot is meant to look like this with the horizontal line.
Can someone please give me some guidance to...
Hi,
I am not sure if I have calculated the task b correctly.
I always interpret an open switch as an infinitely large resistor, which is why no current is flowing through this "resistor". So there is no current in the red circle, as it was the case in task part a, but only in the blue circle...
Hello,
Given a function like ##z= 3x^2 +2y##, the partial derivative of z w.r.t. x is equal to: $$\frac {\partial z}{\partial x} = 6x$$
Let's consider the point ##(3,2)##. If we sat on top of the point ##(3,2)## and looked straight in the positive x-direction, the slope The slope would be...
An infinite product representation of Bessel's function of the first kind is:
$$J_\alpha(z) =\frac{(z/2)^\alpha}{\Gamma(\alpha+1)}\prod_{n=1}^\infty(1-\frac{z^2}{j_{n,\alpha}^2})$$
Here, the ##j_{n,\alpha}## are the various roots of the Bessel functions of the first kind. I found this...
I have been trying to find the financial formula that will give the balance of a credit card debt as a function of time. Example, at 18% interest, if I pay $150 a month how long will it take me to pay off my debt. When I google, I get pointers to Excello functions. I want to know the exact formula.
FIGURE 5 shows an electrically heated oven and its associated control
circuitry. The current, I, to the oven's heating element is fed from a
voltage-controlled power amplifier such that I = EK1. A voltage, VD, derived
from a potentiometer, sets the desired oven temperature, TD. The oven...
As in title:
Plugging in the definition is straight forward, I am too lazy to type, I will just quote the book Fetter 1971:
Up to here everything is very straight forward, in particular, since we are working on free electron gas, ##E=\hbar \omega##
However, I have no idea how to arrive...
Tried to figure out myself but have now admitted defeat, requesting some guidance from you good people. Not looking for any specific answers, unless the problem is my working out and not my process.
If we take the following differential equation: ##y(t)'' + 4y(t) = 7u(t-2)## and determine...
I came across the mentioned equation aftet doing a integral for an area related problem.Doing the maclaurin series expansion for the inverse sine function,I considered the first two terms(as the latter terms involved higher power of the argument divided by factorial of higher numbers),doing so...
An observation I made earlier- something like
def f(...):
...
return ...
def g:
... = f(...)
was quite a bit slower than doing
def g:
f = lambda ... : ...
... = f(...)
any reasons why?
In the picture below we have two identical orbitals A and B and the system has left-right symmetry. I use the notation ##|n_{A \uparrow}, n_{A \downarrow},n_{B \uparrow},n_{B \downarrow}>## which for example ##n_{A \uparrow}## indicates the number of spin-up electrons in the orbital A. I would...
For this problem,
The solution is,
However, I tried to solve this problem using my Graphics Calculator instead of completing the square. I got the zeros of ##x^2 - 2x - 4## to be ##x_1 = 3.236## and ##x_2 = -1.236##
Therefore ##x_1 ≥ 3.236## and ##x_2 ≥ -1.236##
Since ##x_1 > x_2## then...
Let ##f## be a measurable function supported on some ball ##B = B(x,\rho)\subset \mathbb{R}^n##. Show that if ##f \cdot \log(2 + |f|) ## is integrable over ##B##, then the same is true for the Hardy-Littlewood maximal function ##Mf : y \mapsto \sup_{0 < r < \infty}|B(y,r)|^{-1} \int_{B(y,r)}...
My take;
##2\cosh x = e^x +e^{-x}##
I noted that i could multiply both sides by ##e^x## i.e
##e^x⋅2\cosh x = e^x(e^x +e^{-x})##
##e^x⋅2\cosh x = e^{2x}+1##
thus,
##\dfrac{e^x}{1+e^{2x}}=\dfrac{\cosh x + \sinh x}{e^x⋅2\cosh x}##
##= \dfrac{\cosh x +...
I wonder if it's ##f(x)=2^{x}-1## considered an exponential function because in my textbook it's stated that the set of values of an exponential function is a set of positive real numbers, while when graphing this function I get values(y line) that are not positive(graph in attachments), so I am...
This is what I did: $$\lim_ {(x,y) \rightarrow (1,0)} {\frac {g(x)(x-1)^2y}{2(x-1)^4+y^2}}=\lim_ {(x,y) \rightarrow (1,0)} {g(x)y\frac {(x-1)^2}{2(x-1)^4+y^2}}$$ I know that ##\lim_ {(x,y) \rightarrow (1,0)} {g(x)y}=0## and that ##\frac {(x-1)^2}{2(x-1)^4+y^2}## is limited because ##0\leq...
I've already calculated the total spin of the system in the addition basis:
##\ket{1 \frac{3}{2} \frac{3}{2}}; \ket{1 \frac{3}{2} \frac{-3}{2}}; \ket{1 \frac{3}{2} \frac{1}{2}}; \ket{1 \frac{3}{2} \frac{-3}{2}}; \ket{0 \frac{1}{2} \frac{1}{2}}; \ket{0 \frac{1}{2} \frac{-1}{2}}; \ket{1...
I am trying to calculate the partition function of the system of two completely decoupled systems. Probability-wise, the decoupled nature means that the PDF is the product of the PDF of each subsystem. I just wanted to be sure that it would translate into:
$$
H = \sum_{k_i...
I am a bit confused on how we can just say that (z',p) form a 4-vector. In my head, four vectors are sacred objects that are Lorentz covariant, but now we introduced some new variable and say it forms a 4-vector with momentum. I understand that these are just integration variables but I still do...
Preface: I have not done serious math in years. Today I tried to do something fancy for a game mechanic I'm designing.
I've got an item with a variable power level. It uses x amount of ammo to produce f(x) amount of kaboom. Initially it was linear, e.g. fL(x) = x, but I didn't like the scaling...
Hello! I have a plot of a function, obtained numerically, that looks like the red curve in the attached figure. It is hard to tell, but if you zoom in enough, inside the red shaded area you actually have oscillations at a very high frequency, ##\omega_0##. On top of that you have some sort of...
For this problem,
The solution is,
However, I tried solving this problem by using the definition of composite function
##f(g(x)) = f(\frac{4}{3x -2}) = \frac{5}{\frac{4}{3x - 2} - 1} = \frac{5}{\frac{6 - 3x}{3x - 2}} = \frac {15x - 10}{6 - 3x}## which only gives a domain ##x ≠ 2##. Would some...
Hello,
While reading Sakurai (scattering theory/Eikonal approximation section), I encountered a referenced integral
##
\int_0^\infty J_1(x)^2\frac{dx}{x}=1/2
##
I also see this integral from a few places (wolfram, DLMF, etc), so I tried to prove this from various angles (recurrence relations...
This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i##
The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate...
I 've been trying find ##\beta## as a function of ##\theta## for this linkage. It's quite the trigonometric mess.
Start with the Law of Sines:
$$ \frac{\sin \beta}{x} = \frac{\sin \varphi}{R} \implies \boxed{ x = R \frac{\sin \beta}{\sin \varphi} \tag{1} }$$
Relating angles:
$$ \theta +...
Hello all
I am trying to solve the following integral with Mathematica and I'm having some issues with it.
where Jo is a Bessel Function of first kind and order 0. Notice that k is a complex number given by
Where delta is a coefficient.
Due to the complex arguments I'm integrating the...
If f(x) is the function of the "ground": My first assumption is that in a certain $$\bar{x}$$, $$f''(\bar{x})=0$$, and from that point I will analyse the situation.
The object has initial energy $$E_0=\frac{mv^2}{2}+mgf(x),$$ then
$$v=\sqrt{\frac{2}{m}}\sqrt{E_0-mgf(x)}.$$
In each point the...
I know how to solve each of those problems. For the set one, I look at the output of the S and try to match it with the input of T and then take the pair (input_of_S, output_of_T), and I do that for each pair.
As for the formula one, I just plug in x = g(y).
My confusion lies in trying to...