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. SherLOCKed

    A Operation of Hamiltonian roots on wave functions

    How come a+a- ψn = nψn ? This is eq. 2.65 of Griffith, Introduction to Quantum Mechanics, 2e. I followed the previous operation from the following analysis but I cannot get anywhere with this statement. Kindly help me with it. Thank you for your time.
  2. Bunny-chan

    Book demonstration about trigonometric relations

    Homework Statement [/B] In the equation between (3) and (2), why does the author says that ? Isn't the trigonometric identity actually ? 2. Homework Equations The Attempt at a Solution
  3. Mr Davis 97

    I Continuity of composition of continuous functions

    I've learned that composition of continuous functions is continuous. ##\log x## and ##|x|## are continuous functions, but it seems that ##\log |x|## is not continuous. Is this the case?
  4. awholenumber

    B How to Change the Independent Variable from Time to Position in a Function?

    Functions are pretty simple things , they just express a relationship between two different quantities How do i express this function in terms of y(x) = something ?
  5. MAGNIBORO

    I Why are there no other gamma functions?

    hi, i was thinking that every function that satisfies the conditions $$f(0)=1$$ $$f(n+1)=(n+1)f(n)$$ could be a generalization of the factorial function, and why the gamma function is the only function that complies with this conditions? I mean why don't exist other functions, or functions...
  6. F

    Functions and Return statement in C

    Hello, In C (or C++), a function is a body of instructions. Functions can be classified as functions that 1) receive inputs and produce outputs 2) receive no inputs and produce no outputs 3) receive inputs and produce no outputs 4) receive no inputs and produce outputs For case 1) and 4), the...
  7. V

    Bode phase plots and initial angles of transfer functions

    Hello everyone. So I have a test coming up and I am struggling with the concept of figure out what the initial phase or angle of a transfer function is. For instance, consider the following transfer function: L(s) = 4/s(.4s+1)(s+2) So the initial angle for L(s) is -90 degrees. Is there a...
  8. V

    MHB Can We Prove a Function with Intermediate Value Property is Continuous at x?

    Let f be a function with the intermediate value property. In addition, let it have the property that |f(x)-x_n|\le M\cdot sup_{n,m}|f(x_n)-f(x_m)|, where M is a constant and x_n is a sequence converging to x. Then, can we show that f is continuous? I think we have to tackle this problem by...
  9. maistral

    CEOS - Alpha functions requirement

    I seem to have forgotten where I have seen these particular rules; I need them for my research. I think I saw them in a book somewhere... but that was around 4 years ago. Does anyone know where can I find the rules stipulated by statistical mechanics/thermodynamics on the alpha function α(TR)...
  10. S

    A Gauge-invariant operators in correlation functions

    Gauge symmetry is not a symmetry. It is a fake, a redundancy introduced by hand to help us keep track of massless particles in quantum field theory. All physical predictions must be gauge-independent...
  11. binbagsss

    Elliptic Functions, same principal parts, finding additive C

    Homework Statement See attached. The solution of part e) is ##C=4\psi(a)## I am looking at part e, the answer to part d being that the principal parts around the poles ##z=0## and ##z=-a## are the same. Homework EquationsThe Attempt at a Solution [/B] Since we already know the negative...
  12. G

    Expectation values as a phase space average of Wigner functions

    Hi. I'm trying to prove that [\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p) where \rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar}) is the Wigner function, being \rho a density matrix. On the other hand...
  13. binbagsss

    Elliptic functions, removable singularity, limits,

    Homework Statement [/B] please see attached. b) The solution seems a bit vague is the idea here, what this comment is saying, that since this is a simple zero the form of ##lim_{z\to a} f_a(z) (z-a)=0## since, crudely, it is of the form ##\frac{0.0}{0}##. Compared to the point ##z=-a##...
  14. R

    Bounded functions with unbounded integrals

    Homework Statement I am trying to show that the integrator is unstable by giving examples of bounded inputs which produce unbounded outputs (i.e. a bounded function whose integral is unbounded). Note: The integrator is a system which gives an output equal to the anti-derivative of its input...
  15. M

    Moment generating functions help

    Homework Statement [/B] Let X be a random variable with support on the positive integers (1, 2, 3, . . .) and PMF f(x) = C2 ^(-x) . (a) For what value(s) of C is f a valid PMF? (b) Show that the moment generating function of X is m(t) = Ce^t/(2− e^t) , and determine the interval for t for...
  16. B

    A Jacobian Elliptic Functions as Inverse Elliptic Functions

    I need help in understanding how Jacobian Elliptic Functions are interpreted as inverses of Elliptic Functions. Please reference the wiki page on Jacobian Elliptic functions: https://en.wikipedia.org/wiki/Jacobi_elliptic_functions For example, if $$u=u(φ,m)$$ is defined as $$u(φ,m) =...
  17. binbagsss

    Elliptic functions, residue computation, same zeros and poles

    Homework Statement Hi, I am trying to understand the attached: I know that if two functions have zeros and poles at the same point and of the same order then they differ only by a multiplicative constant, so that is fine, as both have a double zero at ##z=w_j/2## and a double pole at...
  18. binbagsss

    Periods of Jacobi Elliptic functions

    Homework Statement I have that ##(\psi(z)-e_j)^{1/2}=e^{\frac{-n_jz}{2}}\frac{\sigma(z+\frac{w_j}{2})}{\sigma(\frac{w_j}{2})\sigma(z)}## has period ##w_i## if ##i=j## and period ##2w_i## if ##i\neq j## where ##i,j=1,2,3## and ##w_3=w_1+w_2## (*) where ##e_j=\psi(\frac{w_j}{2})## I have...
  19. P

    MHB Understanding Sets, Relations, and Functions for Struggling Students

    I'm having issues with the first four questions and have uploaded them. My attempts are shown below. 1. a) True, all elements of E are even b) False, 0 is not a multiple of 3 c) True, 8 is even and 9 is a multiple of 3 d) No idea e) False, 6 is an element of E and T f) No idea 2. a) You can...
  20. FallArk

    MHB Need help, are these functions differentiable?

    I want to figure out whether the functions are differentiable at c. I think I should use some of the trig identities, but I'm not sure which ones. Any tips?
  21. E

    Growth of Functions Homework | Solutions & Analysis

    Homework Statement [/B] Homework Equations Provided in (1). The Attempt at a Solution I think (a) is no because, though ##c_1g > f,## the actual un-vertically-translated ##g## could be less than ##f,## meaning its lower bound ##c_2h < f## over ##c_2 \geq 1,## meaning ##h < f.## Am I...
  22. ShayanJ

    A Integral in terms of Gamma functions

    Does anyone know how I can prove the following equation? ##\displaystyle \frac 1 {d-1}-\int_0^1 \frac{dy}{y^d} \left( \frac 1 {\sqrt{1-y^{2d}} }-1\right)=-\frac{\sqrt \pi \ \Gamma(\frac{1-d}{2d})}{2d \ \Gamma(\frac 1 {2d})} ## Thanks
  23. mabelw

    Deriving functions relating to condition numbers

    I have a question stating to derive the functions x |-> f_1(x)=x^3 and f_2(x)=thirdrootof(x) on their domains of definition based on the asymptotic relative condition number KR = KR(f,x). I'm not sure where to start with this question, I'm not sure if I even understand it. Do I find the...
  24. E

    Algorithm Analysis - Growth of Functions

    The problem statements, all variables and given/known data: Question 1 Question 2 Relevant equations: Provided in question snips. The attempt at a solution: Question 1: I think qux is the answer because it properly increments k by 1 in each iteration. j = j * 2 means j is always 0 so its...
  25. Y

    C/C++ [C++] How to return and call vectors from functions?

    Hi, beginner coder here. I have a somewhat solid understanding of both vectors and functions, and have used the two of them many times, but I'm have trouble coding functions that have vectors in their parameters and as their return values. Another thing I'm having trouble with is calling the...
  26. T

    Injective & Surjective Functions

    Just wondering if anyone could help me get in the right direction with these questions and/or point me to some material that will help me better understand how to approach these questions In what follows I will denote the identity function; i.e. I(x) = x for all x ∈ R. (a) Show that a function...
  27. chakib

    B Sum of increasing and decreasing functions

    i want to know if any real function can be expressed as: f(x)=g(x)+h(x) such as g(x) is an increasing function and h(x) is a decreasing function? thanks
  28. Ben Wilson

    A Coulomb integrals of spherical Bessel functions

    Hi, I'm no expert in math so I'm struggling with solving these integrals, I believe there's an analytical solution (maybe in http://www.hfa1.physics.msstate.edu/046.pdf). $$V_{1234}=\int_{x=0}^{\infty}\int_{y=0}^{\infty}d^3\pmb{x}d^3\pmb{y}\...
  29. binbagsss

    Elliptic functions, periodic lattice, equivalence classes

    Homework Statement ##\Omega = {nw_1+mw_2| m,n \in Z} ## ##z_1 ~ z_2 ## is defined by if ##z_1-z_2 \in \Omega ## My notes say ##z + \Omega## are the cosets/ equivalence classes , denoted by ##[z] = {z+mw_1+nw_2} ## Homework Equations above The Attempt at a Solution So equivalance classe...
  30. A

    I Is there an on-line table of genus of algebraic functions?

    Hi, Given the algebraic function ##w(z)## defined implicitly as ##f(z,w)=a_0(z)+a_1(z)w+a_2(z)w^2+\cdots+a_n(z)w^n=0##, is there any on-line table of genus for them? Haven't been able to find anything. I am writing some code and would like to check it against a standard source. For example...
  31. harpazo

    MHB Extrema of Functions of Two Variables

    Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails. Given: f (x, y) = (x - 1)^2 (y + 4)^2 I found the partial derivative for x and y to be the following: f_x = 2 (x - 1)(y + y)^2 f_y = 2 (y + 4)(x - 1)^2 I solved for x...
  32. Schaus

    Finding discontinuities in functions

    Homework Statement Where are the following functions discontinuous? f(x) = (x+2)/√((x+2)x) Homework EquationsThe Attempt at a Solution f(x) = (x+2)/√((x+2)x) = (x+2)/x√(2x) multiply both denominator and numerator by √(2x) = (x√2+2√x)/(x(2x)) Can I leave it like this and state that x ≠ 0, or...
  33. H

    A Uncertainty Propagation of Complex Functions

    Suppose I have some observables \alpha, \beta, \gamma whose central values and uncertainties \sigma_{\alpha}, \sigma_{\beta}, \sigma_{\gamma} are known. Define a function f(\alpha, \beta, \gamma) which has both real and complex parts. How do I do standard error propagation when imaginary...
  34. J

    Complex periodic functions in a vector space

    Homework Statement Consider the set V + {all periodic *complex* functions of time t with period 1} Draw two example functions that belong to V. Show that if f(t) and g(t) are members of V then so is f(t) + g(t)Homework EquationsThe Attempt at a Solution f(t) = e(i*w0*t)) g(t) =e(i*w0*t...
  35. K

    B Average angle made by a curve with the ##x-axis##

    The average angle made by a curve ##f(x)## between ##x=a## and ##x=b## is: $$\alpha=\frac{\int_a^b\tan^{-1}{(f'(x))}}{b-a}$$ I don't think there should be any questions on that. Since ##f'(x)## is the value of ##\tan{\theta}## at every point, so ##tan^{-1}{(f'(x))}##, should be the angle made by...
  36. K

    I How do I evaluate the Taylor series for ##Si(x)## around a given value?

    Hi, I've got this: $$\sin{(A*B)}\approx \frac{Si(B^2)-Si(A^2)}{2(\ln{B}-ln{A})}$$, whenever the RHS is defined and B is close to A ( I don't know how close). Here ##Si(x)## is the integral of ##\frac{\sin{x}}{x}## But, to check it, I need to evaluate the ##Si(x)## function. I'm new with Taylor...
  37. K

    B Expressions of ##log(a+b), tan^{-1}(a+b),sin^{-1}(a+b)##,etc

    Hi, I got these: $$log(a+b)\approx \frac{b*logb-a*loga}{b-a} + log2 -1$$ $$tan^{-1}(a+b)\approx \frac{b*tan^{-1}2b-a*tan^{-1}2a+\frac{1}{4}*ln\frac{1+4a^2}{1+4b^2}}{b-a}$$ $$sin^{-1}(a+b)\approx \frac{b*sin^{-1}2b-a*sin^{-1}2a+\frac{1}{2}*(\sqrt{1-4b^2}-\sqrt{1-4a^2}}{b-a}$$ And, similarly for...
  38. binbagsss

    Elliptic functions proof -- convergence series on lattice

    Homework Statement Hi I am looking at the proof attached for the theorem attached that: If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2## where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##. For any integer ##r \geq 0 ## : ##\Omega_r := {mw_1+nw_2|m,n \in...
  39. ntran26

    A Inversion of Division of Bessel Functions in Laplace Domain?

    Hello all, I am trying to take the inversion of this function that is in Laplace domain. I've tried using a wolfram alpha solver, and I know I can probably use stehfest algorithm to numerically solve it but wanted to know if there was an exact solution. the function is...
  40. Poetria

    Complex functions with a real variable (graphs)

    Homework Statement How do the values of the following functions move in the complex plane when t (a positive real number) goes to positive infinity? y=t^2 y=1+i*t^2[/B] y=(2+3*i)/t The Attempt at a Solution I thought: y=t^2 - along a part of a line that does not pass through the...
  41. B

    Coarsest Topology With Respect to which Functions are Continuous

    Homework Statement See attached picture.Homework EquationsThe Attempt at a Solution At the moment, I am dealing with part (a). What I am perplexed by is the ordering of the parts. If the subbasis in part (b) does indeed generate this coarsest topology, why wouldn't showing this be included in...
  42. V

    I A few questions about Green's Functions....

    Hi, I am having some trouble understanding exactly when a modified green's function is needed. Here is the general problem: Lu = (p(x)u'(x))' + q(x)u(x) = f(x), x_0 \leq x \leq x_1, p(x) > 0,\\ \alpha_0 u(0) + \beta_0 u'(0) = 0, \alpha_1 u(1) + \beta_1 u'(1) = 0 In my notes it says...
  43. Adolfo Scheidt

    I Product of complex conjugate functions with infinite sums

    Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$ f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) , $$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and...
  44. binbagsss

    Elliptic functions proof - finitely many zeros and poles

    Homework Statement Hi I have questions on the attached lemma and proof. ##f(z)## is an elliptic function here, and non-consant ##\Omega## is a period lattice. So the idea behind the proof is this is a contradiction because the function was assumed to be non-constant but by the theorem that...
  45. K

    B Find ##f(x)## such that f(f(x))=##log_ax##

    I was thinking about extending the definition of superlogarithms. I think maybe that problem can be solved if we find a function ##f## such that ##fof(x)=log_ax##. Is there some way to find such a function? Maybe the taylor series could be of some help. Or is there some method to find a...
  46. LLT71

    I Continous signals as sums of weighted delta functions

    so, continuous signals as sums of weighted delta functions can be represented like this: if you switch order of some variables you get ∫x(τ)δ(-τ+t)dτ, and since,I presume, Dirac delta "function" is even I can write it like this ∫x(τ)δ(-(-τ+t))dτ=∫x(τ)δ(τ-t)dτ=x(t) and we got ourselves a...
  47. L

    Python Recursion in python functions -- confusion

    example code from python-course.eu def factorial(n): print("factorial has been called with n = " + str(n)) if n == 1: return 1 else: res = n * factorial(n-1) print("intermediate result for ", n, " * factorial(" ,n-1, "): ",res) return res...
  48. binbagsss

    A Isomorphism concepts,( example periods elliptic functions )

    Hi, I have the following: Let ##\Omega ## be a discrete subgroup of ##C##, the complex plane. If: i) ##\Omega = \{nw_1 | n \in Z\} ##, then ##\Omega ## is isomorphic to ##Z##. ii) ##\Omega = \{nw_1 + mw_2 | m,n \in Z\} ## where ##w_1/w_2 \notin R ## , then ##\Omega## is isomorphic to ##Z## x...
  49. Austin Chang

    I Understanding Vector Spaces with functions

    Is the set of all differentiable functions ƒ:ℝ→ℝ such that ƒ'(0)=0 is a vector space over ℝ? I was given the answer yes by someone who is better at math than me and he tried to explain it to me, but I don't understand. I am having difficulty trying to conceptualize this idea of vector spaces...
  50. doktorwho

    Examples of functions and sequences

    Homework Statement Give the example and show your understanding: [1][/B].Lets define some property of a point of the function: 1. Point is a stationary point 2. Point is a max/min of a function 3. Point is a turning point of a function If possible name a function whose point has properties of...
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