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.
Humanoid Robots.
Just requiring your thoughts on this.
Major events,functions example Weddings, Birthday, Anniversary celebrations, Cricket, Football live match etc are captured using Video camera/s with Humans performing the function with later on the captured recorded video is edited with...
While deriving Lorentz transformation equations, my professor assumes the following:
As ##\beta \rightarrow 1,##
$$-c^2t^2 + x^2 = k$$
approaches 0. That is, ##-c^2t^2 + x^2 = 0.## But the equation of the hyperbola is preserved in all inertial frames of reference. Why would ##-c^2t^2 + x^2##...
So I thought that the graph tries to tell us that the function is periodic after 2π interval. So I tried to derive its function from the graph as follows using the point slope equation form for the points (0,0) & (a,π): ##y= ({a}/{π})*x##
I hope this function is alright and I just need to find...
Suppose f is a function such that f'(7) is undefined. Which of the following statements is always true? (Give evidences that supports your answer, then explain how those evidences supports your answer)
a. f must be continuous at x = 7.
b. f is definitely not continuous at x = 7.
c. There is not...
I did only the the first three prop.
And on a means we have, on pose or posons means let there be , propriétés means properties, alors meand then.
I apologize i am a french native speaker and i am busy so i cannot rewrite this in entirely english
I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ...
I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof...
I am not sure of the overall purpose of the concepts developed below regarding Riemann integrable functions with compact support ... nor am I sure of the details ... so I am sketching out the meaning as I understand it in 2 dimensions and depicting the relevant entities in diagrams ... I am...
There's a famous functional equation that was asked in the 2019 IMO. It looks like this: find all f: Z -> Z where f(2a)+2f(b)=f(f(a+b)).
I thought of solving it using a recurrence relation where a_n=f(nx). But when I substituted values in the functional equation (after setting a and b equal...
Mathematicians will use the term "elementary functions," often in the context of integration wherein some integrals cannot be expressed in elementary functions.
The elementary functions are usually listed as being arithmetic, rational, polynomial, exponential, logarithmic, trigonometric...
When I look at a range of inputs around x=c and consider the corresponding range of outputs
If 0< |x-c| <δ -> |f(x)-L1|<ϵ1 and |g(x)-L2|<ϵ2 as we shrink the range of inputs the corresponding outputs f(x) and g(x) narrow on L1 and L2 respectively.
|f(x)-L1||g(x)-L2|<ϵ2ϵ1
The product of the...
Can anyone out there give me a hint as to where to start with this problem?
I've been looking at it for a while and can't see a way forward.
What exactly is "the curvature itself" here?BTW I think the dynamic initial value equations 21.116 and 21.117 are incorrect. MTW should have inserted to...
Is it possible for Python matplotlib to plot in one graph a domain dependent function? For example, suppose there is a function where y=x from 0< x <=5, y = x sq when 5<x<=7 and y=2x+9 when x>7. Is it possible in Python to plot this with one plot on one graph? If so, how would it be done?
Suppose f1,f2... is a sequence of functions from a set X to R. This is the set T={x in X: f1(x),... has a limit in R}. I am confused about what is the meaning of the condition in the set. Is the limit a function or a number value? Why?
How can we be sure that a system on the scale of atoms can be described by a single scalar field or the wave function ##\psi##.
I don't just want to do shut up and calculate, maybe using a wave function and then putting it through the time evolution of the Schrödinger equation works, but why...
I want to understand how the domain and range change upon applying transformations like (left/right shifts, up/down shifts, and vertical/horizontal stretching/compression) on functions.
Let f(x)=2-x if 0 ≤x ≤2 and 0 otherwise.
I want to describe the following functions 1) f(-x) 2) -f(x) 3)...
Consider the case of a real function f for which f inverse exists.
1) We we are not used to having the y-axis (vertical axis) to denote the independent variable which it does in x=f-1(y). We rotate the system through positive 90 degree and reflect about the vertical to change the sense of the...
For example:
h(x)=f(x)+g(x)
If f(x) and g(x) are real numbers and real numbers can be added, subtracted, multiplied and divided (except by 0). why do we insist that the x in f(x) and g(x) be {x: x∈ dom f ∩ dom g}?
My thoughts:
The equality of two functions requires two criteria:
1) They operate...
##f## is continuou on ##\mathbb{C}##, so for al ##\epsilon>0##, there is a ##\delta>0## such that $$|\tilde{z}-z|\leq \delta \Rightarrow |f(\tilde{z})-f(z)|\leq \epsilon$$ for all ##\tilde{z}## and ##z## in ##\mathbb{C}##.
Complex conjugation is a norm preserving operation on ##\mathbb{C}##, so...
Problem Statement : Solve for ##x## :
Attempt : If I take ##x=\tan\theta##, the L.H.S. reads $$\tan^{-1}\frac{1-\tan\theta}{1+\tan\theta}= \tan^{-1}\left[\tan\left(\frac{\pi}{4}-\theta \right) \right ]=\frac{\pi}{4}-\theta.$$
On going back to ##x## from ##\theta##, the given equation now...
I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##.
Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...
##lim_{|z|\rightarrow \infty}\frac{f}{g}=1\neq \frac{\infty}{\infty}##
so ##lim_{|z|\rightarrow \infty}f\neq \infty## and ##lim_{|z|\rightarrow \infty}g\neq \infty##.
Because f(z) and g(z) are bounded and entire, f(z) and g(z) are constant functions by Liouville's theorem
.
f(z) and g(z) are...
Hello, I wonder if it is possible to write Bloch wave functions in momentum space.
To be more specific, it would calculate something like (using Sakurai's notation):
$$ \phi(\vec k) = \langle \vec k | \alpha \rangle$$
Moving forward in a few steps:
Expanding:
$$ \phi(\vec k) = \int d^3\vec r...
Hi,
a basic doubt about thermodynamic functions and state variables. Take for instance transformations I and II in the following ##(p,V)## plane.
As far as I can tell, just because the transformations are drawn as continuous lines they are reversible by definition. Namely we can transform...
We show that there is a partition s.t. the upper sum and the lower sum of ##f## w.r.t. this partition converge onto one another.
Let ##\epsilon>0##.
Define a sequence of functions ##g_n:[a,b]\setminus(\{a_n\}_{n\in\mathbb{N}}\cup\{y_0\})## s.t. ##g_n(x)=|f(x)-f(a_n)|##. Suppose there is a...
Hi,
Just curious as to whether distances 'd' , used in Knn ; K nearest neighbors, in Machine Learning, are required to be metrics in the Mathematical Sense, i.e., if they are required to satisfy, in a space A:
##d: A \times A \rightarrow \mathbb R^{+} \cup \{0\} ;
d(a,a)=0 ;
d(a,b)=d(b,a) ...
Consider the following theorem:
Theorem: Let ##f## be a concave differentiable function and let ##g## be a concave function. Then: ##y \in argmax_{x} {f(x)+g(x)}## if and only if ##y \in argmax_{x} {f(y)+f'(y)(x-y)+g(x)}.##
The intuition is that local maxima and global maxima coincide for...
If Tl;dr I am struggling in Math 171 and Physics 191 and throwing around the idea of declaring a geology major with an astronomy minor because the Physics major "juice is not worth the squeeze" at my age(29) anyone else out there who struggled with Calculus 1 when they first took it?Hello...
Let's say I want to describe a massive box in spacetime as described by the Einstein Field Equations. If one were to construct a metric in cartesian coordinates from the Minkowski metric, would it be reasonable to use a piecewise Stress-Energy Tensor to find our metric? (For example, having...
Hi
For a function f ( x , t ) = 6x + g( t ) where g( t ) is an arbitrary function of t ; then is it correct to say that f ( x , t ) is not an explicit function of t ?
For the above function is it also correct that ∂f/∂t = 0 because f is not an explicit function of t ?
Thanks
Hi PF
I have a quote from Spanish 6th edition of "Calculus", by Robert A. Adams, and some queries. I translate it this way:"The inverse of secondary trigonometric functions can easily be calculated with the reciprocal function. For example
DEFINITION 13 The inverse function of secant ##sec^{-1}...
How do I build functions by using Arithmetic Sequence, Geometric Sequence, Harmonic Sequence?
Is it possible to create all the possible function by using these sequences?
Thanks!
Ok in my thinking, i would say that it depends on ##x##, if ##x## belongs to the integer class, then the rational functions would be ##i ## and ##iii##...but from my reading of rational functions, i came up with this finding:
I would appreciate your input on this.
(I must confess that, in spite of working through the chapter on inverse circular functions, I could barely proceed with this problem. Note what it asks to prove : ##x\sqrt{1-x^2}+\ldots## and how much is that at odds with the formula (1 above) of adding two ##sin^{-1}##'s, where you have...
This was the question,
The above solution is the one that I got originally by the question setters,
Below are my attempts (I don't know why is the size of image automatically reduced but hope that its clear enough to understand),
As you can see that both these methods give different answers...
I would like to know if any of you think there's any sort of connection, analogy, or common features between, sets in set theory and wave functions in QT?
Wave functions lack trajectories, so do sets. Wave functions also distribute over areas, as sets can do. To my understanding, wave...
Sinusoidal Functions... Can someone help me with this.
Describe the transformations that are applied to y= -4cos[2(x-30°)] +5 (State any shifts, stretches, compressions, or reflections).
Sometimes, when I code something, I am naming the local variables in the function same as the global variable. Such as,
my_var = 13
def iseven(my_var):
if my_var % 2 == 0:
return True
return False
print(iseven(my_var))
As you can see my_var is defined globally but also used...
I would like to ask whether if operators ##A## and ##B## commute also operators ##e^A## and ##e^B## commute? Also I have a question is it possible that
##e^A## is matrix where all elements are ##\infty## so that ##e^A \cdot e^B-e^B\cdot e^A## has all elements that are ##\infty##?
The following parametrizations assume a counter-clockwise orientation for the unit square; the bounds are ##0\leq t\leq 1##.
Hypotenuse ##(C_1)##
%%%
##r(t)=(1-t,1-t)##
##dr=(-1,-1)\,dt##
##f(r(t))=f(1-t,1-t)=(a(1-t)^2,b(1-t)^2)##
##f\cdot dr=-(a+b)(1-t^2)\,dt##
\begin{align}
\int_{C_1} f\cdot...
DIS observables can be expressed in terms of structure functions F1, F2 and FL. There exists the relation ##F_L = F_2 - 2xF_1##.
We can write $$ F_L = \sum_a x \int_x^1 \frac{dy}{y} C_{a,L}(y,Q) f_a (\frac{x}{y},Q) $$ and similarly for ##F_1## and ##F_2##:
$$ F_1 = \sum_a x \int_x^1...
"Consider a string of length L that is connected at both ends to supports and is subjected to a load (external force per unit length) of f(x). Find the displcament u"
https://i.stack.imgur.com/yVIDG.png
We need to solve this:
$$Tu_{xx} = f(x)$$ subject to $$u(0)=u(L)=0$$
But i don't understand...
Hello everyone first time here. don't know if it's the correct group... Am having some issues wiz my maths homework that going to count as a final assessment. Really Really need help.
The function (f), with a period of 2π is : f(x) = cosh(x-2π) if x [π;3π]..
I had to do a graph as the first...
Hi everyone! =) . I'm having some issues with this exercises, It's about functions. I remember the basic geometrics formulas and how to get the area and perimeter of a square or a circle but I don't get it. I need an explanation.
1. Express the area A of a square as a function of (a) the length...
Problem: Find the cardinality of the set ## A = \{f \in \Bbb N \to \Bbb N. \forall n\leq m .f(n) \geq f (m) \} ##.
I know that ## A \subseteq P(\Bbb N \times \Bbb N) ## implies ## |A| \leq |P(\Bbb N \times \Bbb N)| = | P(\Bbb N) | = \aleph ##. So I have a feeling that ## \aleph \leq |A| ##...
What branch of mathematics studies multinomial functions of matrices? ( i.e matrix valued functions of square matrices such as ##f(A,B,C) = ABC + BAC + 2A^2 + 3C##)