Functions Definition and 1000 Threads

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.

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  1. Ssnow

    Is the Moment Map for a Product Group Action a Morse Function?

    It is know that let ##M## a compact symplectic manifold with ##G=T^{d_{T}}## a torus of dimension ##d_{T}## acting on ##M## in Hamiltonian fashion with Moment map ##\Phi:M\rightarrow \mathfrak{t}^{*}##, then ##\Phi^{\xi}=\langle \Phi(m),\xi\rangle## is a Morse function in each of its component...
  2. Icaro Lorran

    Envelope of a parametric family of functions

    Consider the map ##\phi (t,s) \mapsto (f(t,s),g(t,s))##, a point belonging to the envelope of this map satisfy the condition ##J_{\phi}(t,s)=0##. What is the role of the Jacobian in maps like these and why points in the envelope have to satisfy ##J_{\phi}(t,s)=0##?
  3. D

    Factorization of floor functions of fractions

    hey so if you are taking a floor function of a fraction >1, is there any way to predict anything about it's factorization? what about when the numerator is a factorial and the denominator is made up of factors that divide said factorial but to larger exponents then those that divide the...
  4. F

    Odd/Even functions and integration of them

    I was not sure where to post this here or in calculus, but seeing as the underlying basic principle of my question is regarding parity of functions I am posting it here, but feel free to move if needed. Basically I am getting ready for a (intro to) QM exam and I still struggle with some basic...
  5. G

    MHB Inverse trigonometric functions

    What's $1. ~ \displaystyle \arccos(\cos\frac{4\pi}{3})?$ Is this correct? The range is $[0, \pi]$ so I need to write $\cos\frac{4\pi}{3}$ as $\cos{t}$ where $t$ is in $[0, \pi]$ $\cos(\frac{4\pi}{3}) = \cos(2\pi-\frac{3\pi}{3}) = \cos(\frac{2\pi}{3}) $ so the answer is $\frac{2\pi}{3}$
  6. A

    Integral equivalent to fitting a curve to a sum of functions

    Hello, I am searching for some kind of transform if it is possible, similar to a Fourier transform, but for an arbitrary function. Sort of an inverse convolution but with a kernel that varies in each point. Or, like I say in the title of this topic a sort of continuous equivalent of fitting a...
  7. F

    What Does the Return Statement Indicate in C Functions?

    Hello Forum, I am trying to get clear on the return statement when defining functions in C. A function is a group of statements that together perform a certain task. A function usually receives some input arguments which it uses to produce some output arguments. In C, we must specify what type...
  8. N

    Is a Line Intersecting at One Point a 1-1 Function?

    if we draw a line parallel to the x- axis and passes through a point in the image and the graph intersects at one point is this a one to one function ?
  9. N

    A question on plotting functions on a graph

    when i was reading a supplementary notes doc from open course ware fro MIT on single variable calculus there was a description about a graphical representation of a single valued function as " if each line parallel to the y- axis and which passes through a point in the domain intersects the...
  10. N

    A question on plotting functions on a graph

    when i was reading a supplementary notes doc from open course ware fro MIT on single variable calculus there was a description about a graphical representation of a single valued function as " if each line parallel to the y- axis and which passes through a point in the domain intersects the...
  11. KostasV

    Orthonormality contition for radial functions of hydrogen

    Hello people ! Hope you are fine! I tried to find the inner product that u can see below, between two different radial functions. I was expecting to find zero but i found something nonzero. You can see my two questions below in the photo.
  12. C

    Expressing an integral in terms of gamma functions

    I want to show that $$\int_0^{\infty} \frac{ds}{s-q^2} \frac{s^{-1-\epsilon}}{s-t \frac{z}{1-z}} = \Gamma(1-\epsilon) \Gamma(\epsilon) \frac{1}{t \frac{z}{1-z} - q^2} \left((-t)^{-1-\epsilon} \left(\frac{z}{1-z}\right)^{-1-\epsilon} -(-q^2)^{-1-\epsilon}\right) $$ I have many ideas on how to...
  13. S

    Chain rule for product of functions

    Here is a simple question : let f(g(x)) = h(x)*g(x). I want to calculate df/dx. If I use the product rule, I get g(x)h'(x) + h(x)g'x). Now if I use the composition/chain rule, I get df/dx = df/dg * dg/dx = h(x) * g'(x) which is different. I guess my df/dg = h is wrong, but I can't see what...
  14. H

    Must even functions have even number of nodes?

    The following text considers the possible wave functions when the potential is symmetric about ##x=0##. Why must even functions have an even number of nodes? ##y=sin^2x## is even but always have an odd number of nodes in any interval centred about ##x=0##. The part preceding the above text:
  15. S

    Orthogonality of Wannier functions

    I have trouble reconciling orthogonality condition for Wannier functions using both continuous and discrete k-space. I am using the definition of Wannier function and Bloch function as provided by Wikipedia (https://en.wikipedia.org/wiki/Wannier_function). Wannier function: Bloch function: I...
  16. S

    Integrating Wannier Functions: Simplifying the Prefactor Equation

    Homework Statement I did not manage to get the final form of the equation. My prefactor in the final form always remain quadratic, whereas the solution shows that it is linear, Homework Equations w refers to wannier function, which relates to the Bloch function ##\mathbf{R}## is this case...
  17. chwala

    Differentiation and integration of implicit functions

    1. Given the function ##xy+cos y+6xy^2=0## , it follows that ## dy/dx=-y/x-siny+12xy##2. My problem is how do we integrate this derivative ## dy/dx=-y/x-siny+12xy## to get back the original function3.## ∫dy/dx dx=y ##
  18. ShayanJ

    Are Green's functions generally symmetric?

    In case of the Green's functions for the Laplace equation, we know that they're all symmetric under the exchange of primed and un-primed variables. But is it generally true for the Green's functions of all differential equations? Thanks
  19. saybrook1

    Problem while playing with Bessel functions

    Homework Statement I have run into a number of problems while working through problems regarding Bessel and Modified Bessel Functions. At one point I run into i^{m}e^{\frac{im\pi}{2}} and it needs to equal (-1)^m but I'm not sure how it does. This came up while trying to solve an identity for...
  20. Negatratoron

    Wave Functions: What Are They?

    What's the type of wave functions? Is it just: function from a point in spacetime to Z; takes a location and returns an amplitude in discrete units? (bonus question: according to your favorite theory, what is the type of points in spacetime (that is, topology of spacetime)? is it like r^n for...
  21. M

    MHB Proof that the solutions are algebraic functions

    Hey! :o I am looking at the following: I haven't really understood the proof... Why do we consider the differential equation $y'=P(x)y$ ? (Wondering) Why does the sentence: "If $(3)_{\mathfrak{p}}$ has a solution in $\overline{K}_{\mathfrak{p}}(x)$, then $(3)_{\mathfrak{p}}$...
  22. R

    Evaluating Total Error for Continuous Functions f and g

    Consider two functions f, g that take on values at t=0, t=1, t=2. Then the total error between them is: total error = mod(f(0)-g(0)) + mod(f(1)-g(1)) + mod(f(2)-g(2)) where mod is short for module. This seems reasonable enough. Now, consider the two functions to be continuous on [0,2]. What...
  23. HeavyMetal

    Orthonormal spin functions (Szabo and Ostlund problem 2.1)

    Homework Statement [/B] Taken straight out of Szabo and Ostlund's "Quantum Chemistry" problem 2.1: Given a set of K orthonormal spatial functions, \{\psi_{i}^{\alpha}(\mathbf{r})\}, and another set of K orthonormal functions, \{\psi_{i}^{\beta}(\mathbf{r})\}, such that the first set is not...
  24. little neutrino

    Finding k from Moment Generating Function at t=0

    Homework Statement If M[X(t)] = k (2 + 3e^t)^4 , what is the value of k Homework Equations M[X(t)] = integral ( e^tx * f(x) )dx if X is continuous The Attempt at a Solution I tried differentiating both sides to find f(x), but since it is a definite integral from negative infinity to infinity...
  25. Einj

    Integral of angular functions over d-dim solid angle

    Hello everyone! I have a question about angular integration in arbitrary d dimensions. The interest comes from the need to use dimensional regularization. Suppose I start with a 2-dimensional integral and then I have to move to d=2-\epsilon dimension to regularize my integral. Now, suppose...
  26. J

    Why can no one explain Power Series and Functions clearly

    Hello, Im currently in a Calc II class with unfortunately a bad professor (score of 2 on RateMyProfessor), so I often have to resort to outside sources to learn. Our class is currently on Sequences and Series which has been fine up until we hit the topic of relating Power Series and Functions...
  27. Sollicitans

    Linear Independence of trigonometric functions

    Homework Statement There's no reason to give you the problem from scratch. I just want to show that 5 trigonometric functions are linearly independent to prove what the problem wants. These 5 functions are sin2xcos2x. sin2x, cos2x, sin2x and cos2x. Homework Equations...
  28. C

    System Analysis - Simplifying with Transfer Functions

    Homework Statement Sorry for the pictures, I'd normally write out the problem but it is mostly diagrams. Question and work attached. I am looking for help with part (a) right now, the transfer function I obtain is shown at the end of my work. Homework Equations Knowledge of Laplace transforms...
  29. TheMathNoob

    Functions of two or more random variables

    Homework Statement Supposethat X1and X2 are .random variables and that each of them has the uniform distribution on the interval [0, 1]. Find the p.d.f. of Y =X1+X2. Homework Equations Find cdf of Y and then the pdf The Attempt at a Solution the joint pdf would be f(x1,x2)= 1...
  30. little neutrino

    Statistics - Moment Generating Functions

    If the moment generating function for the random variable X is M[X(t)] = 1/(1+t), what is the third moment of X about the point x = 2? The general formula only states how to find moments about x = 0. Thanks!
  31. terryds

    Inverse and composition of functions

    Homework Statement If ##f(2x-1)= 6x + 15## and ##g(3x+1)=\frac{2x-1}{3x-5}##, then what is ##f^{-1}\circ g^{-1}(3)## ? a) -2 b) -3 c) -4 d) -5 e) -6 The Attempt at a Solution I think the f inverse and g inverse is ##f^{-1}(6x+15)= 2x-1## ##g^{-1}(\frac{2x-1}{3x-5})=3x+1## and,##f^{-1}\circ...
  32. M

    Polygon sine functions? what is this?

    Hi! I was wondering how I could find the equations for the bottom two functions. I understand that the amplitude is not constant like that in the circular sine function--could someone please help me out? Thanks!
  33. W

    Finding the PDF and CDF of a given function Z = X/Y

    Homework Statement Given a Uniform Distribution (0,1) and Z = X/Y Find F(z) and f(z) Homework EquationsThe Attempt at a Solution So I'm just trying to make sure i have the range correct on this one... I'm honestly lost from beginning to end with it. R(z) = {0,∞} because as y is very small, Z...
  34. lordianed

    Prove that no such functions exist

    Homework Statement Prove that there do not exist functions ##f## and ##g## with the following property: $$(\forall x)(\forall y)(f(x+y) = g(x) - y)$$ Homework Equations NA The Attempt at a Solution Here is some information I have found out about ##f## and ##g## if we suppose they exist: ##f(x...
  35. thegirl

    Method used to find harmonic functions in complex analysis

    Hi, I was just wondering how would you go about finding a harmonic function in complex analysis when given certain conditions such as I am z > 0 and is 1 when x > 0 and 0 when x < 0. Do you draw a diagram? Do you solve the laplace equation? How would you go about doing this? What if there...
  36. B

    How to Memorize Even and Odd Functions?

    How do you memorize the even and odd function?
  37. C

    Easiest way to learn exact values for trig functions?

    I'm realizing now how much I need to know the exact values of various trigonometric functions, as shown in various trig tables. Memorizing is pretty arduous, and I'd prefer to understand it, so how can I learn all of these?
  38. C

    Fortran Fortran Functions: How Are Variables Passed Around?

    Hi all Just to start with I should highlight my general ineptitude with programming, so I apologize if my question is totally basic. I have a program written (by someone else) in fortran 77. Contained within it is a call to a function which acts on a number of variables. My curiosity is about...
  39. thegirl

    Finding the limit of lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1

    Hi, I know that when you take this limit it is equal to e^-wo, but I was just wondering how you got there when taking the limit? lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1 = 1/e^wo w and wo are both two points within the same plane.
  40. B

    Exponentials or trig functions for finite square well?

    How do you know when to use exponentials and trig functions when solving for the wave function in a finite square well? I know you can do both, but is there some way to tell before hand which method will make the problem easier? Does it have something to do with parity?
  41. D

    Green's functions for translationally invariant systems

    As I understand it a Green's function ##G(x,y)## for a translationally invariant differential equation satisfies $$G(x+a,y+a)=G(x,y)\qquad\Rightarrow\qquad G(x,y)=G(x-y)$$ (where ##a## is an arbitrary constant shift.) My question is, given such a translationally invariant system, how does one...
  42. Calpalned

    Functions of more than one variable nomenclature

    Homework Statement Homework Equations n/a The Attempt at a Solution ##y'=f(x.y)## is a function of two variables. ##y=y(x)## is a function of only one variable. How can they be related? Clearly ##y(x) = f(x) \neq f(x,y)## Thanks
  43. H

    C program using functions to convert Fahrenheit and Celsius

    Homework Statement Hi i have an assignment that is asking me to convert 3 temperatures in fahrenheit to celcius and vice versa. I am very new to programming only 2 weeks in(and i learn by playing with the program) so I do not know all of the terminology / principles. I am not allowed to use...
  44. Andrew Pierce

    Determining subspaces for all functions in a Vector space

    Homework Statement First, I'd like to say that this question is from an Introductory Linear Algebra course so my knowledge of vector space and subspace is limited. Now onto the question. Q: Which of the following are subspaces of F(-∞,∞)? (a) All functions f in F(-∞,∞) for which f(0) = 0...
  45. evinda

    MHB Find Continuous Functions Subject to an Integral Condition

    Hello! (Wave) I want to find all the continuous functions $f: [0,1] \to \mathbb{R}$ for which it holds that:$$\int_0^1 f(t) \phi''(t) dt=0, \forall \phi \in C_0^{\infty}(0,1)$$ If we knew that $f$ was twice differentiable, we could say that $\int_0^1 f(t) \phi''(t) dt= \int_0^1 f''(t) \phi(t)...
  46. ShayanJ

    Complete set of multi-variable functions

    We know that in the space of functions, its possible to find a complete set so that you can write for an arbitrary function f, ## f(x)=\sum_n a_n \phi_n(x) ## and use the orthonormality relations between ## \phi##s to find the coefficients. But is it possible to find a set of functions ##...
  47. Nemika

    Graphs, equation and functions.

    Is it correct to say that if a relation between a few numbers is represented on a graph and it comes out to be a curve than it can be written in the form of an equation?
  48. C

    MHB Just a question about recursive functions, no code.

    What are different ways of ensuring efficiency in a recursive function in C++? i.e. Prevent calling your recursive function when not necessary.
  49. T

    Derivative of an integral and error functions

    Homework Statement differentiate ∫ e^(-x*t^4)dt from -x to x with respect to x.[/B]Homework Equations erf(x) = (2/sqrt(π)) ∫e^(-t^2)dt from 0 to x. Leibniz rule. I know that ∫t^2e^(-t^2)dt from 0 to x = (√π/4)*erf(x) - (1/2)*x*e^(-x^2)[/B]The Attempt at a Solution By using Leibniz rule...
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