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. K

    Show that the Hydrogen wave functions are normalized

    Homework Statement Show that the (1,0,0) and (2,0,0) wave functions are properly normalized. We know that: Ψ(1,0,0) = (2/(a0^(3/2))*e^(-r/a0)*(1/sqrt(2))*(1/sqrt(2*pi)) where: R(r) = (2/(a0^(3/2))*e^(-r/a0) Θ(θ) = (1/sqrt(2)) Φ(φ) = (1/sqrt(2*pi)) Homework Equations (1) ∫|Ψ|^2 dx = 1 (2)...
  2. R

    Modulus of a complex number with hyperbolic functions

    Homework Statement For the expression $$r = \frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha L)-i\delta\sinh(\alpha L)} \tag{1}$$ Where ##\alpha=\sqrt{\kappa^{2}-\delta^{2}}##, I want to show that: $$\left|r\right|^{2} = \left|\frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha...
  3. I

    B Inner product of functions of continuous variable

    I am new to quantum mechanics and I have recently been reading Shankar's book. It was all good until I reached the idea of representing functions of continouis variable as kets for example |f(x)>. The book just scraped off the definition of inner product in the discrete space case and refined it...
  4. B

    I What are the definitions of Jacobi Elliptic Functions?

    When doing a problem on a pendulum undergoing elliptical motion, I came across sn(z), which is apparently a "Jacobi Elliptic Function". When I looked into it further, I saw that these functions are essentially circular trigonometric functions but about an ellipse instead of a perfect circle. Can...
  5. V

    Fortran Fortran external functions vs subroutines

    Hello, I'm fortran beginner. I have problem with calling or definition of functions and subroutines. I decided to establish my routine file, where I collect all functions and subroutines who I code (for now I have there function for derivative, only this works). My first question, is it good...
  6. Math Amateur

    MHB Analytic Functions - Palka, Ch. III, Section 1.3 ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions ... I need help with fully understanding some remarks by Palka regarding an analytic function in Chapter III, Section 1.3 ... The remarks I refer to from Palka...
  7. Math Amateur

    MHB Continuity of Complex Functions ....Palka, Example 1.5, Chapter 3 .... ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 3: Analytic Functions ... I need help with some aspects of Example 1.5, Chapter 3 ... Example 1.5, Chapter 3 reads as follows: In the above text from Palka Chapter 3, Section 1.2 we...
  8. Math Amateur

    MHB Limits of Complex Functions .... Final Remark from Palka in Section 2.2 ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects regarding an example in Palka's final remarks in Section 2.2 Limits of Functions ... Palka's final remarks in...
  9. Math Amateur

    MHB Limits of Complex Functions .... Example from Palka ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects of a worked example in Palka's remarks in Section 2.2 Limits of Functions ... Palka's remarks in Section 2.2 which...
  10. Math Amateur

    MHB Continuity of Complex Functions .... ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects of the proof of Lemma 2.4 ... Lemma 2.4 and its proof reads as follows: My questions are as follows: Question 1...
  11. RicardoMP

    Mathematica How to append functions to a list of functions?

    My objective is to make a list of functions and afterwards be able to make operations with those functions. Hyfield[list_, bits_] := Module[{i, auxList, Hy}, auxList[a_] := List[]; For[i = 1, i <= bits*2, i++, auxList[a] = Append[auxList[a]...
  12. C

    I Differentiate two variables functions

    Hi, I am dealing with an equality of the form: f(x)=g(y,z) and I need to compute ##dx##. Is the following relation correct? dx={(\frac{\partial f}{\partial x})}^{-1}( \frac{\partial g}{\partial y}dy + \frac{\partial g}{\partial z}dz ) Thank you in advance.
  13. lfdahl

    MHB Determine all real valued differentiable functions f(x)+f(y)=f(xy)

    Determine, with proof, all the real-valued differentiable functions $f$, defined for real $x > 0$, which satisfy $f(x) + f(y) = f(xy)$ for all $x, y > 0$.
  14. C

    Comp Sci Book Database Implementation C++

    I began by creating 2 classes. A book class and a course class that contains any necessary info about the book and course respectively class bookClass{ private: string theISBN; string thebookName; string thebookAuthor; double thebookCost; int...
  15. R

    Exploring Oscillatory Behavior of Sinusoidal Functions

    Homework Statement This isn't really part of my homework, my homework was to draw a pretty graph, but I am curious about some behavior. I was given a picture of a sinusoidal function. I found it was ##2sin(\frac{\pi}{3}t-\frac{\pi}{6}) + 6##. Then I used trig identities to get...
  16. redtree

    B The expectation value of superimposed probability functions

    I apologize for the simplicity of the question (NOT homework). This is a statistical question (not necessarily a quantum mechanical one). If I have an initial probability function with an associated expected value and then a second probability function is superimposed on the initial...
  17. M

    MHB Compositions, Inverses and Combinations of Functions

    HELP! given p(q(x))=2/(5+x) and q(x)=1+x . find a formula for p(x). Someone please help. I don't know how to do this problem .Thanks in advance (PS: would be really helpful if solution is also given)
  18. Math Amateur

    MHB Continuous Functions on Intervals .... B&S Theorem 5.3.2 ....

    I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 5: Continuous Functions ... I need help in fully understanding an aspect of the proof of Theorem 5.3.2 ...Theorem 5.3.2 and its proof ... ... reads as follows:In...
  19. M

    MHB Trigonometry and periodic functions

    !HELP! The Singapore Flyer, until recently the world's largest Ferris wheel, completes one rotation every 32 minutes.1 Measuring 150 m in diameter, the Flyer is set atop a terminal building, with a total height of 165 m from the ground to the top of the wheel. When viewed from Marina Centre, it...
  20. Math Amateur

    MHB Continuous Functions - Thomae's Function ....

    I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 5: Continuous Functions ... I need help in fully understanding an aspect of Example 5.1.6 (h) ...Example 5.1.6 (h) ... ... reads as follows: In the above text from...
  21. Math Amateur

    MHB Limits of Functions .... B&S Theorem 4.2.9 .... ....

    I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 4: Limits ... I need help in fully understanding an aspect of the proof of Theorem 4.2.9 ...Theorem 4.2.9 ... ... reads as...
  22. Math Amateur

    MHB Limits of Functions .... L&S Example 10.7 (2) ....

    I am reading "Real Analysis: Foundations and Functions of One Variable"by Miklos Laczkovich and Vera Sos ... I need help with an aspect of Example 10.7 (2) ... Example 10.7 (2) reads as follows: In the above text, we read the following: "... ... Since whenever \lvert x - 2 \lvert \lt...
  23. Mr Davis 97

    Functions and Analysis with a fixed-point

    Homework Statement A fixed-point of a function f : A → A is a point a ∈ A such that f(a) = a. The diagonal of A × A is the set of all pairs (a, a) in A × A. (a) Show that f : A → A has a fixed-point if and only if the graph of f intersects the diagonal. (b) Prove that every continuous function...
  24. B

    Solutions to Equations Involving Exponential and Trig Functions

    Homework Statement Show that ##e^x = x## does not have any solutions, and show that ##\sec x = e^{-x^2}## has only one solution. Homework EquationsThe Attempt at a Solution Here is my proof of the first proposition: Since ##e^x## is concave up on ##\Bbb{R}##, it must lie above all of its...
  25. karush

    MHB 15.1.55 Find the mass of the plates with the following density functions

    $\textsf{A thin rectangular plate, represented by a region $R$ in the xy-plane}\\$ $\textsf{has a density given by the function p(x,y);}\\$ $\textsf{This function gives the area density in units such as $g/cm^2$}\\$ $\textsf{The mass of the plate is $\displaystyle\iint\limits_{R}p(x,y)dA$}\\$...
  26. S

    Prove transitive (Relations and functions)

    Homework Statement Question 5 of attached photo Homework Equations (a,b)R(c,d) and (c,d)R (e,f) implies (a,b)R(e,f) The Attempt at a Solution Attached photo[/B]
  27. L

    I Can a Circular Function with Complex Variable Represent a 3D Graph?

    Does a circular function with complex variable represent a three-dimensional graph? For example cosiz
  28. Avatrin

    I Rigorously understanding chain rule for sum of functions

    In my quest to understand the Euler-Lagrange equation, I've realized I have to understand the chain rule first. So, here's the issue: We have g(\epsilon) = f(t) + \epsilon h(t). We have to compute \frac{\partial F(g(\epsilon))}{\partial \epsilon}. This is supposed to be equal to \frac{\partial...
  29. ecoo

    Reversed limit definition for monotonic functions

    Homework Statement Does the delta-epsilon limit definition in reverse work for describing limits in monotonic functions? By reversed, one means for lim (x -> a) f(x) = L if for each δ there corresponds ε such that 0 < | x-a | < δ whenever | f(x) - L | < ε. Homework EquationsThe Attempt at...
  30. Oats

    I Must functions really have interval domains for derivatives?

    Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so...
  31. P

    A If [A,B]=0, are they both functions of some other operator?

    In other words, if we are told that A and B commute, then does that mean that there exists some other operator X such that A and B can both be written as power series of X? My instinct is yes but I haven't been able to prove it.
  32. binbagsss

    Elliptic functions: Weierstrass psi function limit

    Homework Statement Show that ##\lim_{z \to 0} z^2( \psi(z)-\psi(\frac{w_j}{2})) =1## where ##\psi(z)=\frac{1}{z^2}+\sum\limits_{w \in \Omega}' \frac{1}{(z-w)^2}-\frac{1}{w^2}## where ##\Omega## are the periods of ##\psi(z)## Homework Equations The Attempt at a Solution ##\lim_{z \to 0}...
  33. M

    What is Implicit Differentiation for a Circle?

    Homework Statement Hello I have this circle with the equation : [/B] (x-a)^2+(y-b)^2=r^2 I want to find dy/dx for it 2. Homework Equations (x-a)^2+(y-b)^2=r^2 The Attempt at a Solution I am looking on the internet and it appears that I should use what is called "Implicit differentiation"...
  34. evinda

    MHB All boolean functions are computed by a depth 2 circuit

    Hello! (Wave) I am looking at the following exercise (pg. 92, ex. 11.1 , book:Gems of Theoretical Computer Science by Uwe Schöning ) : Why can all boolean functions on $n$ variables be computed by a circuit with only 2 levels (a depth 2 circuit) ? What is the size ( number of gates ) of such a...
  35. T

    Increasing and decreasing functions

    Homework Statement f(x)=x[ax-x^2]^ (1/2) for a>0 Then,f(x) A)increases on (3a/4 , a) B)decreases on (0, 3a/4) C)both A,B D)None of these Homework Equations differentiation chain rule f(x) is said to be increasing in (a,b) if it's derivative is positive and decreasing if it's derivative is...
  36. binbagsss

    Elliptic functions proof f(z)-c has N zeros, N the order

    Homework Statement proof of theorem Homework Equations The Attempt at a Solution Hi, I have a couple of questions on the attached proof and theorem 1) On the last line, how is it we go from the order of the zeros = the number of zeros, or is it's meaning the number of zeros counted with...
  37. Telemachus

    Functions of two variables evolving in time with gnuplot

    Hi there. I am making some numerical tests, and printing the results in a data file. The data file contains the Cartesian coordinates, and the function to be plotted at the x,y point for each time t in columns: x,y,f(x,y). I could add a fourth column for the time step, or equally print each time...
  38. J

    B Uniqueness of Analytic Functions

    Hello, I am learning about smooth analytic functions and smooth nonanalytic functions, and I am wondering the following: Is there a theorem that states that for any real analytic functions f and g and a point a, that if at a f=g and all of their derivatives are equal, that then f=g?
  39. Drakkith

    Linear Independence of Two Functions

    Homework Statement Use definition (1) to determine if the functions ##y_1## and ##y_2## are linearly dependent on the interval (0,1). ##y_1(t)=cos(t)sin(t)## ##y_2(t)=sin(t)## Homework Equations (1) A pair of functions is said to be linearly independent on the interval ##I## if and only if...
  40. T

    B Problem solving with hyperbolic functions

    Mod note: Because his caps-lock key is stuck, it's OK for this post to be in all caps. FIRSTLY, MY LAPTOP'S CAPS LOCK IS BEHAVING REALLY WEIRD AND I HAVE NO CONTROL ON IT WHATSOEVER. SO SORRY FOR POSTING IN ALL CAPS/ALL SMALL LETTERS I HAVE RECENTLY LEARNED HYPERBOLIC FUNCTIONS. HOWEVER, I AM...
  41. D

    Problem regarding periodic current functions

    Homework Statement Three periodic currents have the same ##f=100 Hz##. The amplitude of the second current is ##4 A##. and is equal to half of the amplitude of the third current. Effective value of the third current is 5 times that of the first current. At time ##t_1=2ms## third current...
  42. Y

    MHB Partial Order Relation on a Functions Set

    Hello, I have a question which includes several statements, which I need to decide if they are true or false. I am not sure how to do it, if you could give me hints or "leads", it will mostly appreciated. R is a partial order relation on A, a set of functions from [0,1] to [0,infinity) such...
  43. A

    Resonant Frequency and Transfer Functions

    Suppose I have some sort of a filter, whose transfer function is given by H(w), where w is the angular frequency of the input signal in radians per second. I want to know the maximum value of the transfer function. If I solve for the resonant frequency w0, which from my understanding is the...
  44. S

    I Difference between holomoprhic and analytic functions

    Hello! I read in my complex analysis book that holomorphic and analytic "do not always mean the same thing", but in the complex plane they do. In which case they don't mean the same thing? More specifically what does holomoprhic function means outside the complex plane (such that you can define...
  45. R

    Integral simplification using Bessel functions

    Homework Statement I need to simplify the following integral $$f(r, \theta, z) =\frac{1}{j\lambda z} e^{jkr^2/2z} \int^{d/2}_0 \int^{2\pi}_0 \exp \left( -\frac{j2\pi r_0 r}{z\lambda} \cos \theta_0 \right) r_0 \ d\theta_0 dr_0 \tag{1}$$ Using the following integrals: $$\int^{2\pi}_0 \cos (z...
  46. maistral

    Implementing barrier/penalty functions to *Equality* constant

    Hi. I seem to have forgotten how to implement equality constraints to barrier NLPs and quadratic NLPs. Say for example I have this problem: Max Z = x12 + 2 x22 ST: x12 + x22 ≤ 1 x1+ x2 ≤ 1 The unconstrained problem (quadratic penalty - correct me if I'm wrong) then becomes Z = - x12 - 2 x22...
  47. H

    MHB Partial Derivatives of Functions

    I am having some trouble solving the problem shown below. Can anyone point me in the right direction? or provide the location of a worked example? The volume V of a cone of height h and base radius r is given by V=1/3 πr^2 h. The rate of change of its volume V due to stress expansions with...
  48. Wrichik Basu

    A problem in Inverse Circular Functions in Trigonometry

    Homework Statement :[/B] Solve for ##x ##: $$ \sin ^{-1} {x} +\sin ^{-1} {(1-x)} =\cos ^{-1} {x} $$ Answer given: ##0## or ##\frac {1}{2}##. Homework Equations :[/B] All relevant formulae on inverse circular functions may be used. The Attempt at a Solution :[/B] Please see the pic below...
  49. G

    B Add two functions, same frequency to produce one greater?

    Is their any way to add two wave functions like sin or cos in such a way that you could double the frequency or at least increase it?
  50. V

    A Game Theory: Are the payoff functions πi continuous?

    How do I show that the payoff function πi isn´t continuous? Why do best replies not always exist?
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