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. sonnichs

    B Subspaces of Functions- definition

    Assume s is a set such that Fs denotes the set of functions from S-->F where F is a field such as R, C or [0,1] etc. One requirement for F to be a vector space of these functions is closure- e.g. that sums of these functions are in the space: For f,g in Fs the sum f+g must be in Fs hence...
  2. M

    Energy window for wien2wannier

    Hi all, I hope someone can help me with this: I am working with wien2wannier to obtain Wannier functions from wien2k calculations, but I am struggling to define the energy window needed for the init_w2w. The outputfind file shows emin = -999.00 eV, emax = 999.00 eV, and efermi = 0.6124 Ry...
  3. Bling Fizikst

    Uniform Continuity of functions

    Discuss uniform continuity of the following functions: ##\tan x## in ##[0,\frac{\pi}{2})## ##\frac{1}{x}\sin^2 x## in ##(0,\pi]## ##\frac{1}{x-3}## in ##(0,3),(4,\infty),(3,\infty)## I am completely new to this uniform continuity and couldn't find a lot of examples to learn the solving pattern...
  4. imwhatim

    I Analysis of functions of a single variable via multivariable calculus?

    Took calculus of a single variable almost a decade ago where every theorem had to be accepted without proof. Can I fill these gaps by studying a rigorous multivariable/vector analysis book? My justification for this is that R^1 is just a special case of R^n. Or am I looking at this the wrong way...
  5. A

    understanding the options of this website

    assalamualaikum (peace be upon you), brothers and sisters. i m new member. i m actually having trouble with understanding the functions and options of this site. i m unable to find my profile. hope to get help. [Translation by the Mentors via Google Translate] :smile:
  6. tellmesomething

    How to identify the type of function?

    *My attempt:*** We know that $$ (f-g)(x) =\begin{cases} 2x+3&, \ x\in \mathbb{Q}\\ 3x - \sqrt{5} &, \ x\in \mathbb{R} \setminus \mathbb{Q} \end{cases} $$ Now the problem is I don't understand how to navigate it further.. The two equations we obtain would have been straight line equations if...
  7. A

    Bisection and Newton's Approximation

    TL;DR Summary: Writing functions for Bisection and Newtons Approximation in Mathematica Hello! I need to write 2 functions in mathematica, to find the roots of functions. The functions are the Bisection methods and Newtons Approximation. (b1) Write your own function ApproxBisect[a0_,b0_,n_]...
  8. tellmesomething

    Range of a function involving trigonometric functions

    I resolved the numerator to ## cosx-sinx## We get $$mod\frac{cosx-sinx} {3(cosx+sinx)} $$ If we divide the numerator and denominator by cosx we get $$mod\frac{1-tanx} {3(1+tanx)}$$(eq1) We know that tan(π/4-x) is same as ##\frac{1-tanx} {1+tanx}## So re writing eq1 we get $$mod\frac{tan(π/4-x}...
  9. M

    Determining whether functions are sign definite, sign semidefinite, or sign indefinite

    For this problem, I am confused how they get all their negative definite, positive definite, and negative semidefinite domains. I agree that ##V_2## is positive definite by definition. However, for ##V_3## I think they made a mistake since by definition, ##V_3## is negative simi-definite. I'm...
  10. tellmesomething

    Find the range of the function ##f(x) = \frac{e^{2x}-e^x+1} {e^{2x}+e^x+1}##

    My attempt: Say f(x)=y=##\frac{e^{2x}-e^x+1} {e^{2x}+e^x+1}## So I cant rearrange to write in terns of x ##e^{2x}(y-1)+e^x(y+1)+y-1,=0## Now we know that e^x will not be defined for y=1 Therefore when y≠1 ##e^x = \frac{ -y-1+-√((y+1)^2-4(y-1)^2)} {2(y-1)} ## So we can put an inequality on the...
  11. L

    Continuous functions on metric spaces part 2

    Hi, The task is as follows For the proof I wanted to use the boundedness, in the script of my professor the following is given, since both ##(X,d)## and ##\mathbb{R}## are normalized vector spaces I have now proceeded as follows ##|d(x,p)| \le C |x|## according to Archimedes' principle, a...
  12. Rick16

    I How to manipulate functions that are not explicitly given?

    Here are two examples: The first one is from Taylor's Mechanics, section 6.2: Taylor takes the function ##f(y+\alpha\eta, y' + \alpha\eta', x)## and differentiates it with respect to ##\alpha##. In the expression for the function, ##y## and ##\eta## depend on ##x##, ##\alpha## is an...
  13. O

    I Are slices of measurable functions measurable?

    I decided to go through @psie's measure theory notes to refresh myself since it's been a while. I got to the theorem on page 60 which I will attempt to summarize my confusion as just this statement If X and Y are measure spaces and ##f:X\times Y\to \mathbb{C}## is measurable then the function...
  14. M

    Proving rational function converges from first principles

    For this problem, I am confused how they get $$| x - 4 | > \frac{1}{2}$$ from. I have tried deriving that expression from two different methods. Here is the first method: $$-1\frac{1}{2} < x - 4 < -\frac{1}{2}$$ $$1\frac{1}{2} > -(x - 4) > \frac{1}{2}$$ $$|1\frac{1}{2}| > |-(x - 4)| >...
  15. gentzen

    I In which sense(s) do square integrable functions go to zero at infinity?

    In which sense(s) do square integrable functions go to zero at infinity? Of course, they cannot go to zero at infinity in the sense of point evaluation, because point evaluation is not the appropriate concept for square integrable functions. There was a recent discussion in the Quantum Physics...
  16. G

    Finding the constants of the general approximate solution to a double pendulum

    EDIT: My Latex is not showing... Sorry. I attached a file with my "solution". I though this would be quite easy, but I can't seem to solve this system of equations. Should I solve for each mode, or both of them together? I tried to solve them together, here's how far I get: $\text{General...
  17. paulb203

    B Inverse Functions: Why rewrite as y=f(x) ?

    By rewriting, for example, f(x)=2x+3, as y=2x+3, are we simply stating that something = 2x+3; and in the first case we’re calling that something f(x), and in the second case we’re calling it y? Does the y have anything to do with the y axis as in x,y coordinates axes? Or is just a randomly...
  18. chwala

    I Uniform convergence of functions

  19. mathhabibi

    I Alternating Harmonic Numbers are cool, spread the word!

    (Disclaimer: I don't know whether this type of post encouraging discussion on a function is allowed, if not please close this) Hello PF, If you're a fan of integrating, you'll hit a ton of special functions on the way. Things like the Harmonic Numbers, Digamma function, Exponential Integral...
  20. chwala

    Integration of functions of form ##\dfrac{1}{ax+b}##

    This is a bit confusing...conflicting report from attached wolfram and symbolab. Which approach is correct?
  21. chwala

    Solve the given problem involving functions: f(x)=|ax-b| and y=f(x)+c

    A bit confusing here; what i did, Using gradient, we have, ##m=\dfrac{7-1}{0-3}=-2## ##y=-2x+7## Since there is a Vertex, we have the other ##m_2=2## thus, ##y=mx+c## ##1=2×3 +c## ##... y=2x-5## ##a=2, b=-5, c=12## or##a=-2, b=5, c=2##
  22. A

    I Do diffeomorphisms have to be one-to-one functions?

    The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function? Sorry if it's a silly question, I am just a second semester calc student who...
  23. R

    B Functions which relate to calculus: Questions about Notation

    Hi. I'm self-studying functions which relate to calculus. Let me post what I feel I know and what I'm not grasping yet. Please correct any mistakes I'm making. I'm just talking real numbers: A function is a rule that takes an input number and sends it to another number. We can describe it...
  24. M

    Why Is the Chain Rule Not Used in Differentiating h(x) = 3f(x) + 8g(x)?

    For part(a), The solution is, However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule? For example if ##f(x) = \sin(x^2)## Many thanks!
  25. M

    Finding formula for nth derivatives of some functions

    For part(a), The solution is, However, I am having trouble understanding their finial formula. Does anybody please know what the floating ellipses mean? I have only seen ellipses that near the bottom like this ##...## I am also confused where they got the ##2 \cdot 1## from. When solving...
  26. M

    Intro Math Wanting to study Functions from scratch for my Calculus preparation

    Does Precalculus by James Stewart teaches function from scratch ?
  27. T

    A Trig functions and the gyroscope

    Good Morning As I continue to study the gyroscope with Tait-Bryan angles or Euler angles, and work out relationships to develop steady precession, I notice that the trig functions cancel. I stumble on terms like: 1. sin(theta)cos(theta) - cos(theta)sin(theta) 2. Cos_squared +...
  28. J

    A LCAO graphene orbitals wave functions

    Hello, My name is Josip Jakovac, i am a student of the theoretical solid state physics phd studies. First I want to apologize if my question has already been answered somewhere here, I googled around a lot, and found nothing similar. My problem is that I need to apply TBA to Graphene. I went...
  29. Euge

    POTW Analytic Functions with Isolated Zeros of Order k

    Suppose ##f## is analytic in an open set ##\Omega \subset \mathbb{C}##. Let ##z_0\in \mathbb{C}## and ##r > 0## such that the closed disk ##\mathbb{D}_r(z_0) \subset \Omega##. If ##f## has a zero of order ##k## at ##z = z_0## and no other zeros inside ##\mathbb{D}_r(z_0)##, show that there an...
  30. P

    Biology Which organs/parts of the body are only functional on glucose?

    Hi everyone! Do you have an idea which organs/parts of the body are ONLY functional on glucose? I would say the brain, pancreas, liver and kidney, but I have to take into account only those organs that are ONLY functional on glucose
  31. N

    Find an equivalent equation involving trig functions

    Rewrite the given equation, attempt 1: ##2\sin(x)\cos(x) + 2\sin(x) + 2\cos(x) = 0## ##\sin(x)\cos(x) + \sin(x) + \cos(x) = 0## ##\sin(x)(\cos(x) + 1) + \cos(x) = 0##, naaah, can't get any relevant out from here. Attempt 2: ##2\sin(x)\cos(x) + 2\sqrt{2}*\sin(x + \pi/4) = 0## ##\sin(x)\cos(x) +...
  32. P

    C/C++ Overload functions by dimension of vector

    vector<OP> negate (vector<OP> a) { a.insert(a.begin(), neg); return a; } vector<vector<OP>> negate (vector<vector<OP>> a) { for (int i=0; i<a.size(); i++) a[i] = negate(a[i]); // reference to 'negate' is ambiguous? return a; } OP is an enum here. Why can't C++...
  33. M

    I Laplace Transform of Sign() or sgn() functions

    Trying to model friction of a linear motor in the process of creating a state space model of my system. I've found it easy to model friction solely as viscous friction in the form b * x_dot, where b is the coefficient of viscous friction (N/m/s) and x_dot represents the motor linear velocity...
  34. tworitdash

    A Finding Global Minima in Likelihood Functions

    I have a likelihood function that has one global minima, but a lot of local ones too. I attach a figure with the likelihood function in 2D (it has two parameters). I have added a 3D view and a surface view of the likelihood function. I know there are many global optimizers that can be used to...
  35. J

    A An identity with Bessel functions

    Hello. Does anybody know a proof of this formula? $$J_{2}(e)\equiv\frac{1}{e}\sum_{i=1}^{\infty}\frac{J_{i}(i\cdot e)}{i}\cdot\frac{J_{i+1}((i+1)\cdot e)}{i+1}$$with$$0<e<1$$ We ran into this formula in a project, and think that it is correct. It can be checked successfully with numeric...
  36. M

    I Proving Continuous Functions in Smooth Infinitesimal Analysis

    Hello. How to prove that in smooth infinitesimal analysis every function on R is continuous? (Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.) Thanks.
  37. L

    I Limit of the product of these two functions

    If we have two functions ##f(x)## such that ##\lim_{x \to \infty}f(x)=0## and ##g(x)=\sin x## for which ##\lim_{x \to \infty}g(x)## does not exist. Can you send me the Theorem and book where it is clearly written that \lim_{x \to \infty}f(x)g(x)=0 I found that only for sequences, but it should...
  38. Euge

    POTW Uniformly Continuous Functions on the Real Line

    Let ##f : \mathbb{R} \to \mathbb{R}## be a uniformly continuous function. Show that, for some positive constants ##A## and ##B##, we have ##|f(x)| \le A + B|x|## for all ##x\in \mathbb{R}##.
  39. nomadreid

    I Applications of complex gamma (or beta) functions in physics?

    An example of physical applications for the gamma (or beta) function(s) is http://sces.phys.utk.edu/~moreo/mm08/Riddi.pdf (I refer to the beta function related to the gamma function, not the other functions with this name) The applications in Wikipedia...
  40. A

    I What's an example of orthogonal functions? Do these qualify?

    Wiki defines orthogonal functions here https://en.wikipedia.org/wiki/Orthogonal_functions Here's one example, but it's an example that is only true for a specific interval https://www.wolframalpha.com/input?i=integral+sin(x)cos(x)+from+0+to+pi So are these functions orthogonal because there...
  41. A

    I What are the applications of inverses of vector functions?

    As an example, consider a vector-valued function of the form ##f(x,y) = (g_1(x,y),g_2(x,y))##. I typed up one example on wolfram to see if this could be visualized https://www.wolframalpha.com/input?i=plot+f(x,y)+=+(x+y,xy) which was inspired by this question...
  42. A

    I Are there any elementary functions of norms that are still norms?

    If ##d(x,y)## is a metric, then it is said ##\frac{d}{1+d}## is also a metric. I don't know the proof of this, I'd appreciate a reference, but it got me wondering: If ##N(x)## is a norm on a Banach space ##x \in X##, then are there functions in a single real (or complex) variable ##f## (besides...
  43. PhysicsRock

    I Proof about pre-images of functions

    The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##. My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very...
  44. guyvsdcsniper

    Evaluating scalar products of two functions

    I am to consider a basis function ##\phi_i(x)##, where ##\phi_0 = 1 ,\phi_1 = cosx , \phi_2 = cos(2x) ## and where the scalar product in this vector space is defined by ##\braket{f|g} = \int_{0}^{\pi}f^*(x)g(x)dx## The functions are defined by ##f(x) = sin^2(x)+cos(x)+1## and ##g(x) =...
  45. chwala

    Express a function as a sum of even and odd functions

    I am refreshing on this; of course i may need your insight where necessary...I intend to attempt the highlighted...this is a relatively new area to me... For part (a), We shall let ##f(x)=\dfrac{1}{x(2-x)}##, let ##g(x)## be the even function and ##h(x)## be the odd function. It follows...
  46. P

    A Approximating integrals of Bessel functions

    I edited this to remove some details/attempts that I no longer think are correct or helpful. But my core issue is I have never seen this approach to approximating integrals that is used in the attached textbook image. Any more details on what is happening here, or advice on where to learn more...
  47. T

    Strain Energy Functions and Springs

    (If this is in the wrong forum, please move it) Here is the potential energy of a spring Here is the strain energy function in elasticity The look alike -- I like that. If we want the force in the spring, we take the derivative of V with respect to the displacement and make the result...
  48. K

    A Bessel functions of imaginary order

    In Wikipedia article on Bessel functions there is an integral definition of “non-integer order” a (“alpha”). For imaginary order ia I get that Jia* = J-ia, where * is complex conjugate and ia and -ia are subscripts. Then in same article there is a definition of Neumann function, again for...
  49. Mayhem

    I Adding trig functions with different amplitudes

    The trig identities for adding trig functions can be seen: But here the amplitudes are identical (i.e. A = 1). However, what do I do if I have two arbitrary, real amplitudes for each term? How would the identity change? Analysis: If the amplitudes do show up on the RHS, we would expect them...
  50. M

    Python Calling functions without recalling all variables

    Hi PF Given the following def f1(var1, var2): var3 = var1 + var2 return var3 def f2(var1, var2, var4) var4 = 10 var5 = f1(var1, var2)*var4 return var5 it is obvious function f2 does not explicitly need variables var1 and var2. However, it needs the result of f1, which...
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