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.

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  1. Sathish678

    I Independent functions as solutions

    I found three independent functions as solutions for this equation d/dr(r^2dR/dr) = 6R (cauchy equation) r^2 , r^(-3) , (1/7)r^6. But i read that a second order linear differential eqn has only two independent solutions. Why this happened?
  2. SisypheanZealot

    Dirac Delta using periodic functions

    I know it is something simple that I am missing, but for the life of me I am stuck. So, I used the identity ##sin(a)sin(b)+cos(a)cos(b)=cos(a-b)## which gives me $$\int^{\infty}_{-\infty}dx\:f(x)\delta(x-y)=\int^{\infty}_{-\infty}dx\:f(x)\frac{1}{2L}\sum^{\infty}_{n=-\infty}\lbrace...
  3. V

    MHB Computing Kurdyka-Lojasiewicz (KL) exponent of sum of two KL functions.

    Two KL functions $f_1:\mathbb{R}^n\rightarrow \mathbb{R}$ and $f_2:\mathbb{R}^n\rightarrow \mathbb{R}$ are given which have KL exponent $\alpha_1$ and $\alpha_2$. What is the KL exponent of $f_1+f_2$?
  4. D

    MHB How to find functions & inputs whose output is a specific number

    I'm interested in the following problem: given a random number n (n can be gigantic), how do we find a pair function+input(s) whose output is n such that the input(s) are relatively small in size? This problems arises in data compression; consider the bits that make up a file (or a substring of...
  5. B

    I Finding CDF given boundary conditions (simple stats and calc)

    I'm not quite sure if my problem is considered a calculus problem or a statistics problem, but I believe it to be a statistics related problem. Below is a screenshot of what I'm dealing with. For a) I expressed f(t) in terms of parameters p and u, and I got: $$f(t)=\frac{-u \cdot a + u \cdot...
  6. charlesmartin14

    How to compute the surface an N-sphere using delta functions

    Problem Statement: I am trying to understand how to compute the surface an N-sphere , for large N, to leading order (and exactly) Given a vector J with norm N, with N large, how does one compute the volume integral ? That is, what representation of the delta function. And what is the exact...
  7. Z

    Find the functions of V1, V2, V3 in an RC circuit

    I tried using Kirchhof's current law, and to pose the problem in matrix form as ##\frac{dv}{dt}=Mv## with## v## the vector of the ##3## potentials at nodes ##1, 2## and ##3##, and ##M## is a ##3x3## matrix. it would be enough to show me which will be the differential equations, I would proceed...
  8. WMDhamnekar

    MHB Distribution and Density functions of maximum of random variables

    1] Let X,Y,Z be independent, identically distributed random variables, each with density $f(x)=6x^5$ for $0\leq x\leq 1,$ and 0 elsewhere. How to find the distributon and density functions of the maximum of X,Y,Z.2]Let X and Y be independent random variables, each with density $e^{-x},x\geq...
  9. M

    A Proving Lim F(x,y) is the Distribution Function for X

    Let F(x,y) be the joint distribution for random variables X and Y (not necessarily independent). Is ##lim_{y\to \infty}F(x,y)## the distribution function for X? I believe it is. How to prove it?
  10. Q

    Asymptotic behavior of Airy functions in the WKB method

    If it is the asymptotic behavior of the Airy's function what it's used instead of the function itself: Does it mean that the wkb method is only valid for potentials where the regions where ##E<V## and ##E>V## are "wide"?
  11. Benhur

    Combining Sine Functions: Simplifying with Trigonometry

    Moved from technical forum, so no template is shown Summary: I have the expression sin(2x) + sin(2[x + π/3]) and I have to write this in terms of a single function (a single harmonic, rather saying). But I don't know how to do this, and... it seems a little bit weird for me, because I'm merging...
  12. E

    Fourier series for a series of functions

    ## ## Well I start with equation 1): ## e^{b\theta }=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in\theta } ## If ## \theta =0 ## ##e^{b(0)}=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in(0) }## ##1=\frac{sinh(b\pi )}{\pi...
  13. PainterGuy

    H(ω) and H(s) as transfer functions

    Hi, I'm confused between H(ω) and H(s) as transfer functions. The textbook defines both as transfer functions though the term transfer function is mostly reserved for H(s) as far as I can tell. I have read that poles and zeroes of H(s) are helpful in determining the stability. Are poles and...
  14. A

    B Spatial Average of squared functions

    If averaging of a function over a volume is defined as ##\frac{\int_v f(x,y,z,t) dv}{\int_v dv}##. Now if the average ##f^2(x,y,z,t)## is given 0 over a volume,then ##f(x,y,z,t)## has to be necessarily 0 in the volume domain??
  15. S

    Riemann sums for discontinuous functions

    The definition of the Riemann sums: https://en.wikipedia.org/wiki/Riemann_sum I'm stuck with a problem in my textbook involving upper and lower Riemann sums. The first question in the problem asks whether, given a function ##f## defined on ##[a,b]##, the upper and lower Riemann sums for ##f##...
  16. SchroedingersLion

    Integrals over chained functions

    Good evening! Going through a bunch of calculations in Ashcroft's and Mermin's Solid State Physics, I have come across either an error on their part or a missunderstanding on my part. Suppose we have a concatenated function, say the fermi function ##f(\epsilon)## that goes from R to R. We know...
  17. S

    Specifying vertical asymptotes in periodic functions in set notation

    Hi all, What is the general set notation for specifying a vertical asymptote and domain for a periodic function? For example, if I have a periodic function which has a period of pi/2, and within that period, a vertical asymptote occurs at pi/4. The domain is R, excluding that vertical...
  18. Monoxdifly

    MHB [ASK] Proof of Some Quadratic Functions

    So, I found these statements and I need your assistance to prove them since my body condition is not fit enough to think that much. 1. The quadratic equation whose roots are k less than the roots of ax^2+bx+c=0 is a(x+k)^2+b(x+k)+c=0. 2. The quadratic equation whose roots are k more than the...
  19. cookiemnstr510510

    MATLAB logical functions and selection structures

    Hello All, I have a question regarding a MATLAB homework problem. I am learning about logical functions and selection structures. Here is the question: The height of a rocket (in meters) can be represented by the following equation: height=(2.13*t^2)-(0.0013*t^4)+((0.000034*t^(4.751)) create a...
  20. A

    I Single ket for a product of two wave functions

    Hello, I would like to write a product of two wave functions with a single ket. Although it looks simple, I do not remember seeing this in any textbook on quantum mechanics. Assume we have the following: ##\chi(x) = \psi(x)\phi(x) = \langle x | \psi \rangle \langle x | \phi \rangle## I would...
  21. M

    Solve Math Problems: Find Functions Easily

    I really have no clue how to start this. I think I might have to use Pythagoras but I'm really not sure.
  22. Math Amateur

    I Vector Space of Alternating Multilinear Functions ....

    I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ... I need some help in order to fully understand the vector space of alternating multilinear functions ... The relevant text from Shifrin reads as follows: In...
  23. J

    MHB Symmetric/Alternating k-linear functions, Wedge Product

    I am working through Tu's "An Introduction to Manifolds" and am trying to get an understanding of things with some simple examples. The definitions usually seem simple and understandable, but I want to make sure I can use them for an actual function. I've worked a few problems below that my...
  24. Miles123K

    The sum of this series of the product of 2 sine functions

    Homework Statement I have encountered this problem from the book The Physics of Waves and in the end of chapter six, it asks me to prove the following identity as part of the operation to prove that as the limit of ##W## tends to infinity, the series becomes an integral. The series involved is...
  25. M

    A Basis functions and spanning a solution space

    Hi PF Given some linear differential operator ##L##, I'm trying to solve the eigenvalue problem ##L(u) = \lambda u##. Given basis functions, call them ##\phi_i##, I use a variational procedure and the Ritz method to approximate ##\lambda## via the associated weak formulation $$\langle...
  26. M

    Projections of functions and bases

    Homework Statement On ##L_2[0,2\pi]## where ##e = \{ 1/\sqrt{2 \pi},1/\sqrt{\pi}\sin x,1/\sqrt{2 \pi}\cos x \}##. Given ##f(x) = x##, find ##Pr_e f##. Homework Equations See solution. The Attempt at a Solution I take $$e \cdot \int_0^{2\pi} e f(x) \, dx = \pi - 2 \sin x.$$ Look correct?
  27. DuckAmuck

    B Question about how the nabla interacts with wave functions

    Is the following true? ψ*∇^2 ψ = ∇ψ*⋅∇ψ It seems like it should be since you can change the direction of operators.
  28. WMDhamnekar

    MHB What Tension is Required for a 10 kg Mass to Hang Motionless?

    Gravity acting on a 10 kg,mass produces a force of $F_g=\langle 0, -98\rangle$ Newtons. If the mass is suspended from 2 wires which both form $30^\circ$ angles with the horizontal, then what forces of tension are required in order for the mass to hang motionless over time? Answer. I computed...
  29. PhysicS FAN

    Graphs, functions, and coordinates

    Homework Statement If a staight ε: y=(-λ+μ)x +2λ -μ , (where μ and λ are real numbers) passes through point A(0,1) and is parallel to an other straight lin. ζ: y= -2x + 2008 find λ and μ Homework EquationsThe Attempt at a Solution It is clear that when x=0 we know that 2λ-μ=1 which is one of...
  30. W

    MHB Advanced Functions Average vs. Instantaneous velocity

    What do the average velocities on the very short time intervals [2,2.01] and [1.99,2] approximate? What relationship does this suggest exist between a velocity on an interval [a,b] and a velocity near t=a+b/2 for this type of polynomial?
  31. W

    I State functions in Grand Canonical Ensemble vs Canonical

    Hi all, I am slightly confused with regard to some ideas related to the GCE and CE. Assistance is greatly appreciated. Since the GCE's partition function is different from that of the CE's, are all state variables that are derived from the their respective partition functions still equal in...
  32. M

    MHB Bias of functions defined on samples for population

    Let X1, · · · , Xn be a simple random sample from some finite population of values {x1, · · · xN }. Is the estimate \frac{1}{n} \sum_{i}^{n} f(Xi) always unbiased for \frac{1}{N} \sum_{i}^{N} f(xi) no matter what f is?My thinking: I don't think all f's are unbiased, because not all sample...
  33. V

    Find Functions: f o g = Iℝ (ℝ→ℝ)

    Homework Statement Find the functions: f: (0, ∞) → ℝ and g: ℝ → ( 0, ∞) such that f o g = Iℝ (Iℝ denotes identity function on ℝ). Homework EquationsThe Attempt at a Solution I am having trouble working backwards. I know that (f o g)(x) is f(g(x)). I am unsure if this is correct but would f o...
  34. Boltzman Oscillation

    I Are two independent functions constant?

    I was reading Griffith's introduction to QM book and he finds the time independent Schrodinger equation by assuming the wave function to be the product of two independent functions. He eventually gets to this: ih(∂ψ/∂x)/(ψ) = -(h^2/2m)*(∂''φ/∂x^2)/φ + V he says that "the...
  35. M

    MHB Is f in the vector space of cubic spline functions?

    Hey! :o Let $S_{X,3}$ be the vector space of cubic spline functions on $[-1,1]$ in respect to the points $$X=\left \{x_0=-1, x_1=-\frac{1}{2}, x_2=0, x_3=\frac{1}{2}, x_4\right \}$$ I want to check if the function $$f(x)=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |$$ is in $S_{X,3}$...
  36. Boltzman Oscillation

    Square of the sum of two orthonormal functions?

    Homework Statement Given: Ψ and Φ are orthonormal find (Ψ + Φ)^2 Homework Equations None The Attempt at a Solution Since they are orthonormal functions then can i do this? (Ψ + Φ) = (Ψ + Φ)(Ψ* + Φ*)?
  37. Matt Benesi

    B What are cosine and sine functions called in relation to Pi?

    1)* What are sine and cosine functions called in relation to Pi? 2) What is the exponential function called in relation to cosine and sine functions? 3) What are the other smooth, continual nested (or iterative) root functions (that are similar to sine and cosine) called in relation to...
  38. C

    MHB Piecewise Continuous and piecewise smooth functions

    I do not know to start. Here is the problem.Determine if the given function is piecewise continuous, piecewise smooth, or neither. Here $x\neq0$ is in the interval $[-1,1]$ and $f(0)=0$ in all cases. 1. $f(x)=sin(\frac{1}{x})$ 2. $f(x)=xsin(\frac{1}{x})$ 3. $f(x)={x}^{2}sin(\frac{1}{x})$ 4...
  39. C

    MHB Anti-derivatives of the periodic functions

    Dear Everyone, I do not know how to begin with the following problem:Suppose that $f$ is $2\pi$-periodic and let $a$ be a fixed real number. Define $F(x)=\int_{a}^{x} f(t)dt$, for all $x$ . Show that $F$ is $2\pi$-periodic if and only if $\int_{0}^{2\pi}f(t)dt=0$. Thanks, Cbarker1
  40. universal2013

    I Exploring Maxwell Equations & Hyperbolic Functions

    I am trying to understand why maxwell equations are correct in any reference frames? While i started to understand of his laws of physics a bit i could not imagine why he uses hyperbolic functions such as coshw instead of spherical ones in position and time relation between moving frames...
  41. V

    B Linear combination of functions -- meaning?

    h(x) = cf(x) + kg(x) is the linear combination of functions. What makes it linear?
  42. WMDhamnekar

    MHB Joint probability distribution of functions of random variables

    If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$, respectively, compute the joint density of U=X+Y and $V=\frac{X}{X+Y}$ without using Jacobian transformation. Hint:The joint density function can be obtained by differentiating the...
  43. Y

    MHB Quadratic functions: Which of these quadratic functions has exactly one x -intercept

    5. Which of these quadratic functions has exactly one x -intercept? o A. y=x 2 −9 o B. y=x 2 −6x+9 o C. y=x 2 −5x+6 o D. y=x 2 +x−6 A 2. What are the x-intercepts of y=(x−2)(x+5) ? o A. (0, 2) and (0, -5) o B. (0, -2) and (0, 5) o C. (-2, 0) and (5, 0) o D. (2, 0) and (-5, 0) D 5. Which of...
  44. F

    A Wave functions for positrons and electrons

    Is the wave function for the positron the complex conjugate of the wave function for the electron? I've tried to google this, but I can't seem to get a definite answer from a reliable source. It seems that antimatter is derived in quantum field theory which does not concentrate on wave...
  45. T

    Question about the Frobenius method and Bessel functions

    Homework Statement i have been trying to learn bessel function for some time now but to not much help firstly, i don't even understand why frobenius method works why does adding a factor of x^r help to fix the singularity problem. i saw answers on google like as not all function can be...
  46. J

    I Help me understand convolutions and Green's functions

    I'm working through the problems in the first chapter of Jackson and I'm still grappling with the interpretation of Green's functions. I understand that if I have the Poisson equation ##\nabla^2\phi(x) = \frac{-\rho (x)}{\epsilon_0}## and the Green's function ##G(x, x^\prime)## then in general...
  47. D

    Number of functions from a to b where {123} is in the range of (f)

    Homework Statement A has n elements. B={0,1,2,3} {1,2,3}⊆range(f) Homework EquationsThe Attempt at a Solution So in each function we must choose those 3 numbers in the range. So let's first choose all the diffrent possiblites to choose those 3: n*(n-1)*(n-2) now for the remaining elemnts, we...
  48. Prez Cannady

    I Representing nonlinear functions involving vectors

    I'm having trouble finding textbook material on nonlinear functions on vectors. Just as I could define a function ##f## such that: $$f(x) = cos(x)$$ I'd like to write something like: $$f(\vec{x}) = \begin{pmatrix} f_1(x_1) \\ f_2(x_2) \\ ... \\ f_n(x_n) \end{pmatrix} $$ where ##f_i## is...
  49. binbagsss

    Elliptic functions, properties of periods, discrete subgroup

    Homework Statement HiI am following this proof attached and am just stuck on the bit that says: ‘since ##\Omega## is a group it follows that ##|z-\omega|<2\epsilon ## contains..’Tbh, I have little knowledge on groups , it’s not a subject I have really studied in any of my classes-so the only...
  50. binbagsss

    Elliptic functions, diff eq, why proof on open disc holds for C

    Homework Statement Hi I am looking at this derivation of differential equation satisfied by ##\phi(z)##. To start with, I know that such a disc ##D## described in the derivation can always be found because earlier in the lecture notes we proved that their exists an ##inf=min \omega ## for...
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