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.
It is a rather simple question:
In my textbook it writes something like: $$\frac {\partial \Psi} {\partial t}= \frac{i\hbar}{2m}\frac {\partial^2 \Psi} {\partial x^2}- \frac{i}{\hbar}V\Psi$$
$$\frac {\partial \Psi^*} {\partial t}= -\frac{i\hbar}{2m}\frac {\partial^2 \Psi^*} {\partial...
I am trying to call a function declared in a .hpp file and defined in the corresponding .cpp file, from my main.cpp file, but I keep getting an error. From what I have googled it seems as if I am doing this the right way, so I was hoping you guys could help out. Here's my code:
#ifndef CHAP_HPP...
At 02:08, this video shows a function that grows from exactly 0 at input x = 0+, up to 1 at ##x=\infty##.
Its value and all its derivatives approach 0 as x -> 0. The function is Exp(-1 / x^2).
www.youtube.com/watch?v=Wwg_15a0DJo&t=146s
Q. : What function would have its value and all...
Let's consider the Taylor power series of a function on real numbers.
Some of them represent elementary functions, and some of them represent special functions. The special functions cannot be expressed via finite combination of elementary functions on real or complex numbers.
Now, take some...
First thank you for taking your time to take a look at this simple question. And sorry for the informal math language and equations, I hope you guys can understand it.
So, depending on the case, I have 2 or 8 simple quadratic functions f(a), f(b), f(c),… f(z).
Each a,b,c,…,z have a different...
Hi everyone,
We've been looking at Fourier series and related topics in online class, touching upon odd and even periodic extensions. However, we haven't looked at what this translates to for sine and cosine functions - only sawtooth and line examples. So, I'm trying to do my own investigation...
The answer in the textbook writes: $$ f(x) = \frac{1}{4} +\frac{1}{\pi}(\frac{\cos(x)}{1}-\frac{\cos(3x)}{3}+\frac{\cos(5x)}{5} \dots) + \frac{1}{\pi}(\frac{\sin(x)}{1}-\frac{2\sin(2x)}{2}+\frac{\sin(3x)}{3} + \frac{\sin(5x)}{5}\dots)$$
I am ok with the two trigonometric series in the answer...
Hi,
I have a question about a homework problem: I am not sure why I do not seem to get the same answers when using different methods.
Question: Given transfer functions G(s) = \frac{s - 1}{s + 4} and C(s) = \frac{1}{s - 1} , find the state space models for those systems. Then find the...
Here is the solution I have been given:
But I really don't understand this solution. Why can I just add these two exponential factors (adding two individual partition...
For ##x=-1## to be an *horizontal* inflection point, the first derivative ##y'## in ##-1## must be zero; and this gives the first condition: ##a=\frac{2}{3}b##.
Now, I believe I should "use" the second derivative to obtain the second condition to solve the two-variables-system, but how?
Since...
Hello all,
Is this statement true ? Is every increasing monotonic function in a closed interval also continuous ?
How do you prove such a thing ?
Thank you !
Dear all,
I am trying to figure out if a non continuous function is also not bounded. I know that a continuous function in an interval, closed interval, is also bounded. Is a non continuous function in a closed interval not bounded ? I think not, it makes no sense. How do you prove it ?
Thank...
Hi, PF
This is the quote:
"If ##m## is an integer and ##n## is a positive integer, then
6. Limit of a power:
## \displaystyle\lim_{x \to{a}}{\left[f(x)\right]^{m/n}} ## whenever ##L>0## if ##n## is even, and ##L\neq{0}## if ##m<0##"
What do I understand?
-whenever ##L>0## if ##n## is even: ##m##...
Let $a$ and $b$ be two positive integers. Prove that the integer $ a^2+\Bigl\lceil \dfrac{4a^2}{b}\Bigr\rceil$ is not a square.
(Here $\lceil z \rceil$ denotes the least integer greater than or equal to $z$.)
We know that the non-relativistic propagator describes the probability for a particle to go from one spatial point at certain time to a different one at a later time.
I came across an expression (lecture notes) relating ##\Psi(x,t)##, an initial wave function and the propagator. Applying the...
Reading the classical Feynman lectures, I encounter the formula(19.53) that gives the radial component of the wave function:
$$
F_{n,l}(\rho)=\frac{e^{-\alpha\rho}}{\rho}\sum_{k=l+1}^n a_k \rho^k
$$
that, for ##n=l+1## becomes
$$
F_{n,l}=\frac{e^{-\rho/n}}{\rho}a_n\rho^n
$$
To find ##a_n## I...
for reference you can see JS in formulajs.info/functions.
lognormdist use by call : formulajs.LOGNORMDIST(value, mean, stdev, true)
logpearsondist use by call : formulajs.NORMSDIST(z, true)
anybody can help?
Hi all,
I am trying to figure out a way to simplify this problem to give the image of the map. I have not seen this function before and I am having trouble figuring out what the image should come out as.
I have tried graphing u and v separately as a function of x and y in R3 and both surfaces...
Suppose I have a function
$$f(x) = \lim_{\eta \rightarrow 0} \int_{-\infty}^{\infty} d \zeta \frac {g(\zeta)}{x - \zeta + i \eta}$$
and suppose ##g(\zeta)## is a continuous (maybe even differentiable) function. Can ##f(x)## have complex poles of the form ##a + ib## with ##b## not an...
for example, I want to know velocity of a person when time is equal to t, that person start running from 0m/s (t=0s) to max velocity of 1m/s (t=1s). I am thinking that this is like rain droplet that affected by gravity and drag force, where force is directly proportional to its velocity, to make...
I am currently trying to compute the Green's function matrix of an infinite lattice with a periodicity in 1 dimension in the tight binding model. I have matrix ##V## that describes the hopping of electrons within each unit cell, and a matrix ##W## that describes the hopping between unit cells...
The question arose when watching Sean Carroll video: The Biggest Ideas in the Universe _ Q&A 6 - Spacetime 3:50 - 13:30
Because photons follow null geodesic in spacetime the question arose from viewers:
"photons do they really
experience no time this is a question"
And in the answer:
"but if...
Hello fellow physicists,
I need to calculate the rotational partition function for a CO2 molecule. I'm running into problems because I've found examples were they say this rotational partition function is:
##\zeta^r= \frac T {\sigma \theta_r} = \frac {2IkT} {\sigma \hbar^3}##
Where...
Since the index of the root is odd, the domain is going to be ##R##, and I can calculate the second derivative to be:
$$y''=\frac{1}{3}\times \frac{e^x(e^x-3)}{3(e^x-1)^{\frac{5}{3}}}$$
Studying the sign of this function, it results positive for ##x<0 \vee x>ln(3)##, so the main function will be...
I have the solution to the problem, and I mechanically, but not theoretically (basically, why do the C(s) and R(s) disappear?), understand how we go from
##(s^5 + 3s^4 + 2s^3 + 4s^2 + 5s + 2) C(s) = (s^4 + 2s^3 + 5s^2 + s + 1) R(s)##
to
##c^{(5)}(t) + 3c^{(4)}(t) + 2c^{(3)}(t) + 4c^{(2)}(t) +...
is it correct that the continuum states will be free particle states? and the probability will be |< Ψf | ΨB>|^2 . Where Ψf is the wave function for free particle and ΨB is the wave function for the bound state when the depth is B.
Hi,
I'm reading the following paper (L. Chua) about the state-of-art of dynamic non linear circuit analysis -- Chua_Dynamic_Circuits
I've a doubt about Theorem 2 on section 3.2 On the Existence of the Resistor Function that establishes sufficient conditions for the existence of network...
I have the above transfer function for this filter design
The transfer function becomes
which then becomes
Now when I try to use this transfer function to plot a bode plot, I always end up with negative terms which i can't get the square root of
For example, at 1hz, where
w = 6.283
R1 = 1705...
Given the equation : ##|y| x = x##.
Two conditions are possible :
(1) ##\underline{y\geq 0}## : ##xy = x\Rightarrow \boxed{y = 1}\; (x \neq 0)##. We note that except for zero, ##-\infty<x<+\infty## for this case.
(2) ##\underline{y < 0}## : ##-xy = x\Rightarrow \boxed{y = -1}\; (x \neq 0)##...
it really took me time to figure out as to how that equation in aterisk was arrived at...i just noted that both sides of the equation were multiplied by ##-\frac {1}{3}## any particular reason for that? what is the common thinking around that? can one multiply both sides by say ##\frac {1}{6}##...
Hi,
I have this scheme, in which there are 3 segments:
- I is coaxial to c axis and free to rotate in the origin. Length d1
- II is coaxial with a axis and free to rotate around c axis. There a fixed angle θ between a and c axis. Length d2
- III is welded to II, it's the PM segment. α is fixed...
Let's take an example -
f: N-->N
f(x)= 2x
This is called the Rule of the Function.
The rule of the function tells us the relationship between the elements (x,f(x))of the ordered pairs in the Function Set. So, if the...
Considering the measure of angles in radians, that are real numbers, the concept of of trigonometric function spreads to all real numbers. Any real number can be considered as an angle of the first circumference and a ##\mathbb{K}## number of circumferences.
We can consider the function...
$\tiny{gre.al.12}$
The graph of $y=\left|x-6\right|$
is is the standard $(x,y)$ coordinate plane.
Which of the following transformations. when applied to the graph of
$y=\left|x\right|$, in the graph of $y=\left|x-6\right|$?
a. Translation to the right 6 coordinate units
b. Translation to...
When studying classical mechanics we are told that light is the propogation of electromagnetic waves. This makes perfect sense, as I can imagine these fields behaving this way, and in turn have an associated wave length. When learning about QM, I have heard that the wavelength of a (any)...
I have a project folder called Projects and inside that folder I have a file called my_functions.py. Now I have a folder named Project 1
and inside there is another python file called test.pySo it is something like
-Projects-
my_functions.py
-Project 1-
test.py
Now I...
I calculated the derivative of this function as:
$$\frac{6x^3-4x}{3\sqrt[3]{(x^3-x)^2}}$$
Now, in order to find and later study non-differentiable points, I must find the values which make the argument of the root equal to zero:
$$x^3-x=0 \rightarrow x=0 \vee x=\pm 1$$
and then find the left and...
I am struggling to understand Callen's explanation for stability, I understand that the concavity of S(U) must be negative because otherwise we can show that this means that the temperature increases as the internal energy decreases (dT/dU<0) but I cannot understand equation (8.1) which...
In the notation of Function ---> f(x)=y
Here f represents the function and x is the variable in the function.
we read f(x) as "f of x"or "function of the variable x"
what does "function of x " mean that we read ?
in this notation what does round brackets ()...
Hi! I have a question about integrating Planck's function of blackbody radiation. Why is it that the area under the blackbody curve will be less than the spectral radiance of individual wavelengths? For example, integrating the Sun's curve over all wavelengths yields a smaller value than the...
I have to find:
g(1)=
and
g(5)=
I have drawn the graph and I am a little unsure where to go from there. I know area is involved somehow but not entirely sure what to do. Any help is appreciated
I try to proof it but i got stuck right here, i want your opinions
Can I get a solution if i continue by this way? or Do I have to take another? and if it is so, what would yo do?
Let a continuous function f: R -> R such that f(x)f(f(x)) = 1 and f(2020) = 2019. What is the value of f(2018)?
I am having problem with this question.
I already tried, through various attempts, to find the function explicitly which satisfy this condition and f(0) different of 0. But it leads...