What is Functions: Definition and 1000 Discussions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.
For this problem,
I am confused how they get $$| x - 4 | > \frac{1}{2}$$ from. I have tried deriving that expression from two different methods. Here is the first method:
$$-1\frac{1}{2} < x - 4 < -\frac{1}{2}$$
$$1\frac{1}{2} > -(x - 4) > \frac{1}{2}$$
$$|1\frac{1}{2}| > |-(x - 4)| >...
In which sense(s) do square integrable functions go to zero at infinity?
Of course, they cannot go to zero at infinity in the sense of point evaluation, because point evaluation is not the appropriate concept for square integrable functions. There was a recent discussion in the Quantum Physics...
EDIT: My Latex is not showing... Sorry. I attached a file with my "solution".
I though this would be quite easy, but I can't seem to solve this system of equations. Should I solve for each mode, or both of them together? I tried to solve them together, here's how far I get:
$\text{General...
By rewriting, for example, f(x)=2x+3, as y=2x+3, are we simply stating that something = 2x+3; and in the first case we’re calling that something f(x), and in the second case we’re calling it y?
Does the y have anything to do with the y axis as in x,y coordinates axes? Or is just a randomly...
(Disclaimer: I don't know whether this type of post encouraging discussion on a function is allowed, if not please close this)
Hello PF,
If you're a fan of integrating, you'll hit a ton of special functions on the way. Things like the Harmonic Numbers, Digamma function, Exponential Integral...
A bit confusing here; what i did,
Using gradient, we have,
##m=\dfrac{7-1}{0-3}=-2##
##y=-2x+7##
Since there is a Vertex, we have the other ##m_2=2##
thus,
##y=mx+c##
##1=2×3 +c##
##... y=2x-5##
##a=2, b=-5, c=12## or##a=-2, b=5, c=2##
The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function?
Sorry if it's a silly question, I am just a second semester calc student who...
Hi. I'm self-studying functions which relate to calculus. Let me post what I feel I know and what I'm not grasping yet. Please correct any mistakes I'm making.
I'm just talking real numbers: A function is a rule that takes an input number and sends it to another number. We can describe it...
For part(a),
The solution is,
However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule? For example if ##f(x) = \sin(x^2)##
Many thanks!
For part(a),
The solution is,
However, I am having trouble understanding their finial formula. Does anybody please know what the floating ellipses mean? I have only seen ellipses that near the bottom like this ##...## I am also confused where they got the ##2 \cdot 1## from.
When solving...
Good Morning
As I continue to study the gyroscope with Tait-Bryan angles or Euler angles, and work out relationships to develop steady precession, I notice that the trig functions cancel.
I stumble on terms like:
1. sin(theta)cos(theta) - cos(theta)sin(theta)
2. Cos_squared +...
Hello,
My name is Josip Jakovac, i am a student of the theoretical solid state physics phd studies.
First I want to apologize if my question has already been answered somewhere here, I googled around a lot, and found nothing similar.
My problem is that I need to apply TBA to Graphene. I went...
Suppose ##f## is analytic in an open set ##\Omega \subset \mathbb{C}##. Let ##z_0\in \mathbb{C}## and ##r > 0## such that the closed disk ##\mathbb{D}_r(z_0) \subset \Omega##. If ##f## has a zero of order ##k## at ##z = z_0## and no other zeros inside ##\mathbb{D}_r(z_0)##, show that there an...
Hi everyone!
Do you have an idea which organs/parts of the body are ONLY functional on glucose?
I would say the brain, pancreas, liver and kidney, but I have to take into account only those organs that are ONLY functional on glucose
vector<OP> negate (vector<OP> a) {
a.insert(a.begin(), neg);
return a;
}
vector<vector<OP>> negate (vector<vector<OP>> a) {
for (int i=0; i<a.size(); i++)
a[i] = negate(a[i]); // reference to 'negate' is ambiguous?
return a;
}
OP is an enum here. Why can't C++...
Trying to model friction of a linear motor in the process of creating a state space model of my system. I've found it easy to model friction solely as viscous friction in the form b * x_dot, where b is the coefficient of viscous friction (N/m/s) and x_dot represents the motor linear velocity...
I have a likelihood function that has one global minima, but a lot of local ones too. I attach a figure with the likelihood function in 2D (it has two parameters). I have added a 3D view and a surface view of the likelihood function. I know there are many global optimizers that can be used to...
Hello.
Does anybody know a proof of this formula?
$$J_{2}(e)\equiv\frac{1}{e}\sum_{i=1}^{\infty}\frac{J_{i}(i\cdot e)}{i}\cdot\frac{J_{i+1}((i+1)\cdot e)}{i+1}$$with$$0<e<1$$
We ran into this formula in a project, and think that it is correct. It can be checked successfully with numeric...
Hello.
How to prove that in smooth infinitesimal analysis every function on R is continuous? (Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.)
Thanks.
If we have two functions ##f(x)## such that ##\lim_{x \to \infty}f(x)=0## and ##g(x)=\sin x## for which ##\lim_{x \to \infty}g(x)## does not exist. Can you send me the Theorem and book where it is clearly written that
\lim_{x \to \infty}f(x)g(x)=0
I found that only for sequences, but it should...
Let ##f : \mathbb{R} \to \mathbb{R}## be a uniformly continuous function. Show that, for some positive constants ##A## and ##B##, we have ##|f(x)| \le A + B|x|## for all ##x\in \mathbb{R}##.
An example of physical applications for the gamma (or beta) function(s) is
http://sces.phys.utk.edu/~moreo/mm08/Riddi.pdf
(I refer to the beta function related to the gamma function, not the other functions with this name)
The applications in Wikipedia...
Wiki defines orthogonal functions here
https://en.wikipedia.org/wiki/Orthogonal_functions
Here's one example, but it's an example that is only true for a specific interval
https://www.wolframalpha.com/input?i=integral+sin(x)cos(x)+from+0+to+pi
So are these functions orthogonal because there...
As an example, consider a vector-valued function of the form ##f(x,y) = (g_1(x,y),g_2(x,y))##.
I typed up one example on wolfram to see if this could be visualized
https://www.wolframalpha.com/input?i=plot+f(x,y)+=+(x+y,xy)
which was inspired by this question...
If ##d(x,y)## is a metric, then it is said ##\frac{d}{1+d}## is also a metric. I don't know the proof of this, I'd appreciate a reference, but it got me wondering:
If ##N(x)## is a norm on a Banach space ##x \in X##, then are there functions in a single real (or complex) variable ##f## (besides...
The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##.
My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very...
I am to consider a basis function ##\phi_i(x)##, where ##\phi_0 = 1 ,\phi_1 = cosx , \phi_2 = cos(2x) ## and where the scalar product in this vector space is defined by ##\braket{f|g} = \int_{0}^{\pi}f^*(x)g(x)dx##
The functions are defined by ##f(x) = sin^2(x)+cos(x)+1## and ##g(x) =...
I am refreshing on this; of course i may need your insight where necessary...I intend to attempt the highlighted...this is a relatively new area to me...
For part (a),
We shall let ##f(x)=\dfrac{1}{x(2-x)}##, let ##g(x)## be the even function and ##h(x)## be the odd function. It follows...
I edited this to remove some details/attempts that I no longer think are correct or helpful.
But my core issue is I have never seen this approach to approximating integrals that is used in the attached textbook image. Any more details on what is happening here, or advice on where to learn more...
(If this is in the wrong forum, please move it)
Here is the potential energy of a spring
Here is the strain energy function in elasticity
The look alike -- I like that.
If we want the force in the spring, we take the derivative of V with respect to the displacement and make the result...
In Wikipedia article on Bessel functions there is an integral definition of “non-integer order” a (“alpha”). For imaginary order ia I get that Jia* = J-ia, where * is complex conjugate and ia and -ia are subscripts. Then in same article there is a definition of Neumann function, again for...
The trig identities for adding trig functions can be seen:
But here the amplitudes are identical (i.e. A = 1). However, what do I do if I have two arbitrary, real amplitudes for each term? How would the identity change?
Analysis: If the amplitudes do show up on the RHS, we would expect them...
Hi PF
Given the following
def f1(var1, var2):
var3 = var1 + var2
return var3
def f2(var1, var2, var4)
var4 = 10
var5 = f1(var1, var2)*var4
return var5
it is obvious function f2 does not explicitly need variables var1 and var2. However, it needs the result of f1, which...
Hello. I am reading about state equations from a physics textbook, Physics by Frederick J. Keller, W. Edward Gettys, Malcolm j. Skove (Volume I). I don't understand some parts but since I have the Turkish translation of the book I must translate it as good and clear as possible.
"State...
Let ##\epsilon>0##. Choose ##N\in\mathbb{N}## s.t. for each integer ##n## s.t. ##n\geq N##,
$$|\sup\{|f-f_n|(x):x\in D\}|<\frac{\epsilon}{3}$$
where ##D## denotes the intersection of the domains of ##f## and ##f_n##.
Choose a partition ##P:=\{x_0,\ldots,x_m\}## with ##x_i<x_{i+1}## where...
Hi, I'm writting because I sort of had an idea that looks that it should work but, I did not find any paper talking about it. I was thinking about approximating something like algebraic functions. That is to say, a function of a complex variable z,(probably multivalued) that obeys something...
##f : [0,2] \to R##. ##f## is continuous and is defined as follows:
$$
f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$
$$
f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$
##V = \text{space of all such f}##
What would the basis for V? Well, for ##x \in [0,1]## the basis for ##V##...
I argue not. Let ##f:\mathbb{Q}\rightarrow\mathbb{R}## be defined s.t. ##f(r)=r^2##. Consider an increasing sequence of points, to be denoted as ##r_n##, that converges to ##\sqrt2##. It should be clear that ##\sqrt2\equiv\sup\{r_n\}_{n\in\mathbb{N}}##. Continuity defined in terms of sequences...
So I have $$f(x,y,z,t,n) = 0,g(x,y,z,t,n) = 0,h(x,y,z,t,n) = 0 $$ and i need to find the best ##[x,y,z,t]## that fit the data, where n is the variable. Now, the amount of data for each function is pretty low (2 pair for f (that is, two (n,f)), 3 pair for g and another 3 pair for h)
The main...
Hello
Simple question
Whether the minimum of the product of two functions in one single variable, is it greater or less than the product of their minimum
thanks
Sarrah
R is the triangle which area is enclosed by the line x=2, y=0 and y=x.
Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v##
## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as...
From what I've gather the primary benefits to worm wheels are:
- their ability to provide high reduction ratios
- self-locking which can be useful for hoisting and lifting applications.
- Operates silently and smoothly, which reduces vibrations
Feel free to add any important ones I might've...
Because it drives to contradictions. Here is a nice example from E. Rosinger Generalized solutions of nonlinear PDE.
We can multiply generalized functions from ##\mathcal D'(\mathbb{R})## by functions from ##C^\infty(\mathbb{R})##. This operation is well defined. For example $$x\delta(x)=0\in...
Hello, recently I'm learning about correlation functions in the context of QFT. Correct me with I'm wrong but what i understand is that tha n-point correlation functions kinda of describe particles that are transitioning from a point in space-time to another by excitations on the field. So, what...
As is well known, almost periodic functions can be represented as a Fourier series with incommensurable (non-multiple) frequencies https://en.wikipedia.org/wiki/Almost_periodic_function. It seems to me that I came up with an integral criterion for the degree of non-periodicity. The integral of a...
*** MENTOR NOTE: This thread was moved from another forum to this forum hence no homework template.
Summary:: Trying to find transfer functions to design a block diagram on simulink with a PID controller and transfer functions for a water tank system.
----EDIT---
The variables and parameters...