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.
Homework Statement
Determine whether or not the following sequences of real valued functions are Cauchy in L^{1}[0,1]:
(a) f_{n}(x) = \begin{cases} \frac{1}{\sqrt{x}} & , \frac{1}{n+1}\leq x \leq 1 \\ 0 & , \text{ otherwise } \end{cases}
(b)
f_{n}(x) = \begin{cases} \frac{1}{x} & ...
Y=f(x)
which passes through points:
(-1,3) and (0,2) and (1,0) and (2,1) and (3,5)
second function is defined: g(x)=2f(x-1)
Calculate g(0)
Calculate g(1)
Calculate g(2)
Calculate g(3)
NOTE:This is not a homework question! This is just a topic that I like very much,but don’t have the programming ability to do many of them.That’s why I post this thread.
C++ is a language without built-in big integer calculation functions,so building ones that can do such job is a great way to...
Homework Statement
I am suppose to write a program that compares the FFT (Fast Fourier Transform Diagrams) of a sampled signal without the use of a window function and with it. The window function should be as long as the signal and the signal should have N points, N chosen as to not cause...
I'm trying to prove that the set of all square integrable functions f(x) for which ∫ab |f(x)|^2 dx is finite is a vector space. Everything but the proof of closure is trivial.
To prove closure, obviously we should expand out |f(x)+g(x)|^2, which turns our integral into one of |f(x)|^2 (finite)...
I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard.
I am currently focused on Section 3.1: Manifolds ...
I need some help in order to understand Example 3.1.3 ... ...
Example 3.1.3 reads as follows:In...
$$\int x^2+3 = \frac{x^3}{3}+3x+C$$
I can get the front two part by power rule, but what is the C doing there? Wolframalpha suggested it should be a constant, but what value should it be? Sorry for asking rookie questions:-p
I have the statement \sin[\sin^{-1}(x)] = x \hspace{7pt} if -1 \leq x \leq 1. How can I tell if plugging in x will return x for \cos[\cos^{-1}(x)] and \tan[\tan^{-1}(x)] ? What if the positions of the regular and inverse functions were reversed? For example, \cos^{-1}[\cos(x)].
I am only...
If you would allow me to ask...
if i have two convex functions , and i was to place one inside the other, i.e. convolute them...what could be said in general about the resultant function.
what information about the original functions can be taken from the positions of the minima.
and is there...
Definition: A function f mapping from the topological space X to the topological space Y is continuous if the inverse image of every open set in Y is an open set in X.
The book I'm reading (Charles Nash: Topology and Geometry for Physicists) emphasizes that inversing this definition would not...
Homework Statement
Define {x \choose n}=\frac{x(x-1)(x-2)...(x-n+1)}{n!} for positive integer n. For what values of positive integers n and m is g(x)={{{x+1} \choose n} \choose {m}}-{{{x} \choose n} \choose {m}} a factor of f(x)={{{x+1} \choose n} \choose {m}}?
Homework Equations
The idea...
Hello
I have tried to resolve an exercise which is asking how the graph is modified according to the variables into the function. I would appreciate any help since accordin to my udnerstanding the function should increase
Please, follow below:
Suppose y0 is the y-coordinate of the point of...
I have a set of values and I'm trying to come up with functions to fit that data.
Here is what I know about the data:
It is rounded down / floored to the nearest significant digit (i.e. 1 for v1 and v3, 0.1 for v2).
Columns v1 and v3 look linear (e.g. first order polynomial).
Column v2 looks...
In analysis, the pasting or gluing lemma, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. Can we have a similar situation for uniform continuous functions?
Let R\subseteq A*B be a binary relation from A to B , show that R is a function if and only if R^-1(not) R \subseteq idB and Rnot aR^-1 \supseteq both hold. Remember that Ida(idB) denotes the identity relation/ Function {(a.a)|a A} over A ( respectively ,B)
Please see the attachment ,I...
Let f be a function from a set X to a set Y, moreover, A ⊆ X. What comparison sign canput instead? to assert "f ^(−1) (f(A)) ? A" become true? (Possible signs of comparison in this : ⊆, ⊇, =. It is necessary to take into account all options.
f ^(−1) - inverse of fall options.), Let f be a...
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...
I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ...
I need help in order to fully understand Theorem 12.7, Section 12.9 ...
Theorem 12.7...
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ...I need help in order to fully understand Theorem 12.7, Section 12.9 ...Theorem 12.7...
Homework Statement
Let ##F## be an entire function such that ##\exists## positve constants ##c## and ##d## where ##\vert f(z)\vert \leq c+d\vert z\vert^n, \forall z\in \Bbb{C}##.
Is this question incomplete? My complex analysis course is not rigorous at all and this came up on a past final...
In Theodore Shifrin's book: Multivariable Mathematics, he defines the derivative of a multivariable vector-valued function as follows:
Lafontaine in his book: An Introduction to Differential Manifolds, defines the derivative of a multivariable vector-valued function slightly differently as...
Homework Statement
Let R be the area in the xy-plane in the 1st quadrant which is bounded by the curves y^2+x^2 = 5, y = 2x and x = 0. (y-axis). Let T be the volume of revolution that appears when R is rotated around the Y axis. Find the volume of T.
Homework EquationsThe Attempt at a Solution...
Consider the following limit where L'H Rule was correctly applied twice
Determine the functions f'(x), g'(x), f(x), and g(x) needed to result in the limit given.
\begin{align*}\displaystyle
\lim_{x \to 0}\frac{f(x)}{g(x)}
\overset{\text{L'H}}=&
\lim_{x \to...
Justify the following by using table, graph and equation. use words to explain each representation
f(X) = 2 x2 - 8x and g(x) = x2-3x+ 6 the points (-1,10) and (6,24)
Sorry for all the questions. Reviewing for my midterm next week. Fun fun.
If someone could take a look at my proof for (a) and help me out with (b) that'd be awesome!
(a) Let $\Delta$ be a partition of $[a, b]$ that is a refinement of partition $\Delta'$. For a real-value function $f$ on $[a...
Homework Statement
This is a translation so sorry in advance if there are funky words in here[/B]
f: ℝ→ℝ a function 2 time differentiable on ℝ. The second derivative f'' is bounded on ℝ.
Show that the sequence on functions $$ n[f(x + 1/n) - f(x)] $$ converges uniformly on f'(x) on ℝ...
Define $f(x)=sinx$ on $[0, 2\pi]$. Find two increasing functions h and g for which f = h−g
on $[0, 2\pi]$.
I know that if f is of bounded variation in $[a,b]$, it is the difference of two positive, monotonic increasing functions. However, we didn't do any examples of this in class. Is there a...
Homework Statement
Let ##X \subset \mathbb{C}##, and let ##f_n : X \rightarrow \mathbb{C}## be a sequence of functions. Show if ##f_n## is uniformly Cauchy, then ##f_n## converges uniformly to some ##f: X \rightarrow \mathbb{C}##.
Homework Equations
Uniform convergence: for all ##\varepsilon >...
Hi PF!
The ODE $$g''(x) + (1-k^2)g(x) = f(x)\\ g(0) = y(\pi/3) = 0$$
where ##f(x)## is a forcing function and ##k \in \mathbb N## is a constant has a Green's function via variation of parameters as
$$
G_L = \frac{L(y)R(x)}{W} : 0<x<y<\pi/3\\
G_R = \frac{L(x)R(y)}{W} : 0<y<x<\pi/3
$$
with...
Hello folks,
I'm glad that I discovered this forum. :) You might save me.
I'm hearing right now differential geometry and am having some problems with the subject.
May you explain me the follwoing. We had the special case of the i-th projection. My lecturer now posited that the differential of...
Hi PF!
I am trying to solve an ODE by casting it as an operator problem, say ##K[y(x)] = \lambda M[y(x)]##, where ##y## is a trial function, ##x## is the independent variable, ##\lambda## is the eigenvalue, and ##K,M## are linear differential operators. For this particular problem, it's easier...
The answer for derivative of y=5tanx+4cotx is y'=-5cscx^2. But how come on math help the answer is 5sec^2x-4csc^2x? I have a calculus test coming up and I really would appreciate if someone could explain!
- - - Updated - - -
Oh nvm I see my mistake!
Let a function ##f:X \to X## be defined.
Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##.
Then which of the following are correct ?
a) ##f(A \cup B) = f(A) \cup f(B)##
b) ##f(A \cap B) = f(A) \cap f(B)##
c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)##
d) ##f^{-1}(A \cap...
Homework Statement
f(x)= x/(1+x)
What is f(f(x)) and what is its domain.
2. The attempt at a solution
I found f(f(x))= x/(1+2x)
and the domain: (-∞,-1/2)∪(-1/2,∞) , but it is saying that I have the wrong domain. What mistake have I made?
My process for finding domain:
1. Find the domain...
Please see my attached image, which is a screenshot from Khan Academy on the limits of composite functions.
I just want to check if I'm understanding this correctly, particularly for #1, which has work shown on the picture.
Now my question:
We are taking the limit of a composition of...
Hi members,
I have a problem with the computation of residues involve the gamma functions.
(see attached Pdf file)
Can you show me for the first residue with the arrows, or give a hint or a link.
Thank you
Hi everyone,
So I am a high school student and I am learning calculus by myself right now (pretty new to that stuff still). Currently I am working through some problems where integration leads to logarithm functions. While doing one of the exercises I noticed one thing I don't understand. I...
One of our homework problem asks:
If f is a one-to-one function such that f(-3)=5 , find x given that f^-1 (5)=3x-1.
Here's how I attempted to solve the problem:
-3=3x-1
3x=-2
x=-2/3
Is this the correct way to solve the problem?
Hello,
I am reading the book QFT for the gifted amateur and I have a question concerning how to go from the wave function picture to the Green's function as defined by equations (16.13) and (16.18) at page 147.
## \phi(x,t_{x}) = \int dy G^{+}(x,t_{x},y,t_{y})\phi(y,t_{y}) ##...
Homework Statement
Let A, B, C be finite sets such that A and B have the same number of elements, that is, |A| = |B|. Let f : A → B and g : B → C be functions.
(a) Suppose f is one-to-one. Show that f is onto.
(b) Suppose g ◦ f is one-to-one. Show that g is one-to-one.Homework EquationsThe...
Hi All
I normally post on the QM forum but also have done quite a bit of programming and did study computer science at uni. I have been reading a book about Ramanujan and interestingly he was also good friends with Bertrand Russell. You normally associate Russell with philosophy but in fact...
Homework Statement
1.1.3
1) Do functions that vanish at the endpoints x=0 and L=0 form a vector space?
2) How about periodic functions? obeying f(0)=f(L) ?
3) How about functions that obey f(0)=4 ?
If the functions do not qualify, list what go wrong.Homework Equations
The Attempt at a...
Homework Statement
Q1. a) In relation to Fourier analysis state the meaning and significance of
4
i) odd and even functions ii) half-wave symmetry {i.e. f(t+π)= −f(t)}.
Illustrate each answer with a suitable waveform sketch.
b) State by inspection (i.e. without performing any formal analysis)...
In the first volume of his lectures (cap. 6-5) Feynman asserts that these 2 can be the PDF of velocity and position of a particle.
Under which conditions it's possible to model velocity and position of a particle using these particular PDFs ?
ps: Is the "Heisenberg uncertainty principle"...
1. Determine the equation that represents the relationship between the power and the current when the electric potential difference is 24v and the resistance is 1.5 Ω. 2. Draw a graph of the parabola that corresponds to the equation found in (a). 3. Determine the current needed in order for...
Homework Statement
The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong.
Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...
Hello!
I'd like to ask for a help about how to compute accurately functions which has very intense oscillations. My example is to estimate
I = \int_0^{\infty} \sin(x^2) dx= \int_0^{\infty}\frac{\sin(t)}{2\sqrt{t}} dt.I tried trapezoid rule over one oscillation at a time, but result is poor. My...
Hi PF!
I'm working with some basis functions ##\phi_i(x)##, and they get out of control big, approximately ##O(\sinh(12 j))## for the ##jth## function. What I am doing is forcing the functions to zero at approximately 3 and 3.27. I've attached a graph so you can see. Looks good, but in fact...
Hi PF!
I have a function that looks like this $$f(r,\theta) = \sinh (\omega \log (r))\cos(\omega(\theta - \beta))$$
You'll notice ##f## is harmonic and satisfies the BC's ##f_\theta(\theta = \pm \beta) = 0##. Essentially ##f## has no flux into the wall defined at ##\theta = \pm \beta##. So we...