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. ognik

    MHB Reverse direction for complex functions

    Hi An exercise asks to show $ \int_{a}^{b}f(z) \,dz = -\int_{b}^{a}f(z) \,dz $ I can remember this for real functions, something like $ G(x) = \int_{a}^{b}f(x) \,dx = G(b) - G(a), \therefore \int_{b}^{a}f(x) \,dx = G(a) - G(b) = -\int_{a}^{b}f(x) \,dx $ I have seen 2 approaches, either...
  2. Amrator

    Rate of Change Using Inverse Trig Functions

    Homework Statement A spectator is standing 50 ft from the freight elevator shaft of a building which is under construction. The elevator is ascending at a constant rate of 20 ft/sec. How fast is the angle of elevation of the spectator's line of sight to the elevator increasing when the elevator...
  3. W

    How to Generate a 3D Grid for Tight Binding Wave Functions?

    Dear all, Could somebody please, indicate me some tutorial, in order to generate a 3D grid to plot the wave function using the Hamiltonian eigenvalues and the slater type orbitals ? Thanks in advance, Wellery
  4. C

    Writing correct mathematics -- functions within functions....

    Hi I'm a bit confused about some mathematical notation If i write f(x)=(2x^2 + 10)^4 And i define u= 2x^2 +10 u^4 = f(x) Would it then be correct to write f(u)= u^4 Or would i get f(u)= 2(u)^2 +10 = (2(2x^2 +10)+10)^2 Should i define u^4 = f(x) first? Would it then be correct...
  5. F

    Linear Algebra vector functions LI or LD

    Homework Statement Determine whether or not the vector functions are linearly dependent? u=(2t-1,-t) , v= (-t+1,2t) and they are written as columns matrixes. Homework Equations Wronskian, but I don't think I should use it because I need to take derivatives so it doesn't seem like it would...
  6. ognik

    MHB Do Cauchy-Riemann Conditions Guarantee Analyticity?

    Hi - just started complex analysis for the 1st time. I have been a little confused as to the chicken and egg-ness of Cauchy-Riemann conditions... 1) Wiki says: "Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy–Riemann...
  7. C

    Mixing units with functions or derivatives?

    Hi, How do you correctly use units when writing derivatives and functions in math? Example A car goes 17miles per gallon, so a function m with the equation m(g)=17g describes the distance it can go with g gallons. And the derivative dm/dg = 17 miles/gallon. Question: could you write the...
  8. E

    Re-scaling Functions under the Same Axes

    Consider two functions ##f\left(x, y\right)## and ##g\left(px, qy\right)##, where ##p## and ##q## are known. How can I plot the two functions on the same graph (i.e. the same axes)? The function ##f\left(x, y\right)## will have axes with values ##x## and ##y##, while the other will have axes...
  9. D

    Sum of Related Periodic Functions

    I have been looking through the book Counterexamples: From Elementary Calculus to the Beginning of Calculus and became interested in the section on periodic functions. I thought of the following question: Suppose you have a periodic real valued function f(x) with a fundamental period T. Let c...
  10. Math Amateur

    MHB Polynomials and Polynomial Functions in I_m = Z/mZ

    I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra. I am currently focused on Section 3. Polynomials I need help with the a statement of Rotman's concerning the polynomial functions of a finite ring such as \mathbb{I}_m = \mathbb{Z}/ m \mathbb{Z} The relevant section...
  11. Essence

    Java Defining variables in the context of implicit functions in java

    Sorry for the disturbance, So I have been looking (without success) for a way to define a variable within an implicit function in Java. What I really mean by this is I have the equation: In this function my program will give me all of the values except for px. I have tried rearranging the...
  12. C

    Do wave functions go to zero at ~ct?

    Suppose you have a free election and you make a measurement of its position r_0 at time t = 0. You then wait some time t required for the wave function to evolve out of its collapsed eigenstate. The electron now supposedly has a wave function expanding to infinity in all directions, albeit with...
  13. M

    Motivation of sin and cos functions

    Is there a way to motivate the sinus and cosinus functions by looking at their Taylor expansion? Or equivalently, is there a way to see that complex numbers adds their angles when multiplied without knowledge of sin and cos?
  14. N

    Proof using hyperbolic trig functions and complex variables

    1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a)Homework Equations I have derived. tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y) tan(2x) = 2tanx / 1 - tan^2 (x) Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B]3. My attempt: By...
  15. D

    Sum of Two Periodic Orthogonal Functions

    Homework Statement This problem is not from a textbook, it is something I have been thinking about after watching some lectures on Fourier series, the Fourier transform, and the Laplace transform. Suppose you have a real valued periodic function f with fundamental period R and a real valued...
  16. J

    Are functions partly defined by their domains and codomains?

    I just finished working through compositions of functions, and what properties the inner and outer functions need to have in order for the whole composition to be injective or surjective. I checked Wikipedia just to make sure I'm right in thinking that for a composition to be injective or...
  17. S

    Irrational Roots Theorems for Polynomial Functions

    Is any Irrational Roots Theorem been developed for polynomial functions in the same way as Rational Roots Theorems for polynomial functions? We can choose several possible RATIONAL roots to test when we have polynomial functions; but if there are suspected IRRATIONAL roots, can they be found...
  18. W

    Integrals of the Bessel functions of the first kind

    Hi Physics Forums. I am wondering if I can be so lucky that any of you would know, if these two functions -- defined by the bellow integrals -- have a "name"/are well known. I have sporadically sought through the entire Abramowitz and Stegun without any luck. f(x,a) = \int_0^\infty\frac{t\cdot...
  19. O

    MHB Decreasing nonnegative sequence and nonincreasing functions

    Let $\{p_n\}$ be a nonnegative nonincreasing sequence and converges to $p \ge 0$. Let $f : [0,\infty)\to[0,\infty)$ be a nondecreasing function. So, since f is a nondecreasing function, $f(p_n)>f(p)>0$. How did this happen?
  20. P

    Inverse trigonometric functions

    I am familiar with the importance of the following inverse circular/hyperbolic functions: ##\sin^{-1}##, ##\cos^{-1}##, ##\tan^{-1}##, ##\sinh^{-1}##, ##\cosh^{-1}##, ##\tanh^{-1}##. However, I don't really get the point of ##\csc^{-1}##, ##\coth^{-1}##, and so on. Given any equation of the form...
  21. S

    A difficult problem on functions

    Homework Statement I've been trying to solve the following problem but can't wrap my head around it. Let ##x##, ##f(x)##, ##a##, ##b## be positive integers. Furthermore, if ##a > b##, then ##f(a) > f(b)##. Now, if ##f(f(x)) = x^2 + 2##, then what is ##f(3)##? Homework Equations The Attempt...
  22. P

    Inverse hyperbolic functions (logarithmic form)

    To express the ##\cosh^{-1}## function as a logarithm, we start by defining the variables ##x## and ##y## as follows: $$y = \cosh^{-1}{x}$$ $$x = \cosh{y}$$ Where ##y ∈ [0, \infty)## and ##x ∈ [1, \infty)##. Using the definition of the hyperbolic cosine function, rearranging, and multiplying...
  23. B

    How to correctly convolve two delta functions?

    How do I correctly compute the convolution of two delta functions? For example, if I want to compute ##\delta(\omega)\otimes\delta(\omega)##, I should integrate $$\int_{-\infty}^\infty \delta(\omega-\Omega)\delta(\Omega) d\Omega$$ This integrand "fires" at two places: ##\Omega = 0## and...
  24. A

    Transforming Functions: Solving g(x) = 2f(-x+(3/2))

    Homework Statement If f(x)=|x-1/2|-5 determine g(x)=2f(-x+(3/2)) Homework Equations The Attempt at a Solution Well, I tried to factor out the k-value in the g(x) formula. So I was left with: g(x)=2f(-1)(x-3/2) Then I multiply f(x) by 2 and am left with: g(x)=2|x-(1/2)|-10 Then I subtract...
  25. RyanH42

    Geometry Vector Functions Searching Video Lecture

    Hi I am studying Differantial Geometry.My textbook is Lipschultz Differantia Geometry and I am in chapter two.I undersand the basic idea of Vector function but I wanI to gain more information.Is there any video lectures about vector functions.I found this...
  26. D

    Arguments of exponential and trig functions

    What can be said about the arguments of the exponential functions and trig functions ? Can the argument be a vector or must it be a scalar ? If it can only be a scalar must it be dimensionless ?
  27. S

    Optimization methods with bivariate functions

    Hi, I have the following equation: f(z)=g(z)+b*u(z) where z=(x,y) i.e. bivariate,b is a parameter, u(z) the uniform distribution and g(z) a function that represents distance. By considering for a momment b=0, min(f(z)) can give me the location of the minimum distance. However because I want...
  28. Jaco Viljoen

    Let f, g and h be functions defined below:

    Homework Statement f(x)=(√x^2-3x+2)/(2x-3), g(x)=3/(√(x+3)) and h(x)=(x^2-5x+6)/(x-2) which of the following are true: A)Df={x∈ℝ:x≤1 or x≥2} B)Dg={x∈ℝ:x≥-3} C)Dh=ℝ Homework EquationsThe Attempt at a Solution I am using substitution here by replacing the x by the parameters specified and...
  29. Jaco Viljoen

    Let f, g and h be functions defined as follows:

    Homework Statement f(x)=(√x2-3x+2)/(2x-3), g(x)=3/(√x+3) and h(x)=(x2-5x+6)/(x-2) which of the following are true: A)Df={x∈ℝ:x≤1 or x≥2} B)Dg={x∈ℝ:x≥-3} C)Dh=ℝ Homework EquationsThe Attempt at a Solution I am only attempting now,
  30. S

    Modeling Functions.... real life business, money, probability, etc.

    Ok. I'm now studying economics and applied math, and I'm currently wanting to know what book or online resource could help me in learning how to model real life situations into functions. Most math and econ textbooks are garbage at this. I'll be more specific. In my study of Microeconomics...
  31. rolotomassi

    Green's Functions and Feynman Diagrams

    I've been learning about Greens functions. I'm familiar with how to find them for different differential operators and situations but far from fully understanding them. We were shown in lecture how they can be used to obtain a perturbation series, leading to Feynman diagrams which represent...
  32. Rafael de Gomes

    Maple - Finding maximum of maximums in different functions

    Hello, I've been trying to come up with a short way of writing the code. What I'm trying to do is: I have 11 equations, each of which have a defined minimum and maximum. I'm trying to find the highest maximum out of all of them and I need to know which one it is. The highest as in farthest...
  33. T

    MHB Solve Multiline Function w/2 Variables - Explanation Here

    Can anyone explain how to solve a multiline function with two variables? Please see attached.
  34. V

    Find Intersections of Trig Functions with different periods

    There are 2 trig functions on the same set of axis. f(x)=600sin(2π3(x−0.25))+1000 and f(x)=600sin(2π7(x))+500 How do I go about finding the points of intersections of the two graphs? This was from a test I had recently and didn't do too well on,so any help would be much appreciated. I started...
  35. evinda

    MHB Are the Weak and Strong Norms Correct for These Function Distances?

    Hello! (Wave) In $C^1[0,1]$ calculate the distances between the functions $y_1(x)=0$ and $y_2(x)=\frac{1}{100} \sin(1000x)$ in respect to the weak and strong norm.That's what I have tried:Weak norm: $||y_1-y_2||_w=\max_{0 \leq x \leq 1} |y_1(x)-y_2(x)|+ \max_{0 \leq x \leq 1} |y_1'(x)-y_2'(x)...
  36. S

    Square-integrable functions as a vector space

    Homework Statement (a) Show that the set of all square-integrable functions is a vector space. Is the set of all normalised functions a vector space? (b) Show that the integral ##\int^{a}_{b} f(x)^{*} g(x) dx## satisfies the conditions for an inner product. Homework Equations The main...
  37. K

    Where does band index come from in block wave functions?

    Can you show me as explicitly as possible through equations?
  38. Kyuutoryuu

    Are hyperbolic functions used in Calculus 3?

    More than just a few problems that happen to pop up in the textbook, I mean.
  39. K

    Finding poles of complex functions

    I am trying to calculate a pole of f(z)=http://www4b.wolframalpha.com/Calculate/MSP/MSP86721gicihdh283d613000033ch4ae4eh37cbd4?MSPStoreType=image/gif&s=35&w=44.&h=40. . The answer in the textbook is: Simple pole at...
  40. Chrono G. Xay

    'Wheel-like' Mathematics (Modulating Trig Functions?)

    As part of a personal musicology project I found myself with the mathematical model of a geometry which utilizes the equation a*(a/b)sin(pi*x) The only problem with this is that I need to take the integral from -1/2 <= x <= 1/2, and according to Wolfram Alpha no such integral exists. I can...
  41. A

    Why do collapse functions change form in master equations?

    In the GRW spontaneous collapse model (for example) the wave-function evolves by linear Schrödinger equation, except, at random times, wave-function experiences a jump of the form: \psi_t(x_1, x_2, ..., x_n) \rightarrow \frac{L_n(x)\psi_t(x_1, x_2, ..., x_n)}{||\psi_t(x_1, x_2, ..., x_n)||}...
  42. K

    Why does an inverse exist only for surjective functions?

    In other words, why does a function have to be onto (or surjective, i.e. Range=Codomain) for its inverse to exist?
  43. M

    MHB Can We Enumerate All Primitive Recursive Functions?

    Hey! :o Can we enumerate the primitive recursive functions?
  44. S

    Orthogonality relations for Hankel functions

    Where can I find and how can I derive the orthogonality relations for Hankel's functions defined as follows: H^{(1)}_{m}(z) \equiv J_{n}(z) +i Y_{n}(z) H^{(2)}_{m}(z) \equiv J_{n}(z) - i Y_{n}(z) Any help is greatly appreciated. Thanks
  45. gonadas91

    Can We Determine a Complex Function from Its Poles Alone?

    Suppose we have a complex function f(z) with simple poles on the complex plane, and we know exactly where these poles are located (but we don't know how the function depends on z) Is there any way to build up the exact form of f(z) just from its poles?
  46. H

    Raabe's test says Legendre functions always converge?

    The Legendre functions are the solutions to the Legendre differential equation. They are given as a power series by the recursive formula (link [1] given below): ##\begin{align}y(x)=\sum_{n=0}^\infty a_n x^n\end{align}## ##\begin{align}a_{n+2}=-\frac{(l+n+1)(l-n)}{(n+1)(n+2)}a_n\end{align}##...
  47. ChrisVer

    Positive integral for two functions

    I am feeling stupid today...so: Is the following statement true? \int_{-\infty}^{+\infty} f(x) g(x) \ge 0 if f(x) \ge 0 and 0< \int_{-\infty}^{+\infty} g(x) < \infty (integral converges)? If yes, then how could I show that? (It's not a homework) I am trying to understand how the...
  48. gonadas91

    Meromorphic functions expansions

    Hi, is there any good book or table with all the known expansions for meromorphic functions in the complex plane¿ (Using Mittag-Leffer theorem to express the function as a sum of its poles) I am trying to evaluate an infinite sum which seems rather complicated and I wonder if there is something...
  49. P

    Delay complexity of Boolean functions

    Homework Statement I don't really understand how/why every Boolean function of n variables may be implemented with a delay complexity of O(n). Could someone please try and explain? Homework EquationsThe Attempt at a Solution I was trying to show it using minterms (SOP). There is a maximum of...
  50. A

    Greens functions from path integral

    Let me post this question again in a slightly modified form. On the attached picture the path integral for the partion function: Z = Tr(exp(-βH)) Now according to what it says on the picture it should be easy from this to get the Green's function in the path integral formalism. The Green's...
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