What is Functional: Definition and 414 Discussions

In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that map values to other values, rather than a sequence of imperative statements which update the running state of the program.
In functional programming, functions are treated as first-class citizens, meaning that they can be bound to names (including local identifiers), passed as arguments, and returned from other functions, just as any other data type can. This allows programs to be written in a declarative and composable style, where small functions are combined in a modular manner.
Functional programming is sometimes treated as synonymous with purely functional programming, a subset of functional programming which treats all functions as deterministic mathematical functions, or pure functions. When a pure function is called with some given arguments, it will always return the same result, and cannot be affected by any mutable state or other side effects. This is in contrast with impure procedures, common in imperative programming, which can have side effects (such as modifying the program's state or taking input from a user). Proponents of purely functional programming claim that by restricting side effects, programs can have fewer bugs, be easier to debug and test, and be more suited to formal verification.Functional programming has its roots in academia, evolving from the lambda calculus, a formal system of computation based only on functions. Functional programming has historically been less popular than imperative programming, but many functional languages are seeing use today in industry and education, including Common Lisp, Scheme, Clojure, Wolfram Language, Racket, Erlang, Elixir, OCaml, Haskell, and F#. Functional programming is also key to some languages that have found success in specific domains, like JavaScript in the Web, R in statistics, J, K and Q in financial analysis, and XQuery/XSLT for XML. Domain-specific declarative languages like SQL and Lex/Yacc use some elements of functional programming, such as not allowing mutable values. In addition, many other programming languages support programming in a functional style or have implemented features from functional programming, such as C++11, Kotlin, Perl, PHP, Python, Go, Rust, Raku, Scala, and Java (since Java 8).

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  1. V

    What is generating functional and vacuum-to-vacuum boundary conditions in QFT?

    Hello everyone :) I'm reading the book QFT - L. H. Ryder, and I don't understand clearly what are the generating functional Z[J] and vacuum-to-vacuum boundary conditions? Help me, please >"<
  2. Pythagorean

    Transforming association matrix to a functional matrix

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  3. J

    Proving Symmetric Operators are Equal: A Functional Analysis Challenge

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  4. D

    What is the proof for the functional derivative formula in DFT?

    Hi, in their book ''Density-Functional Theory of Atoms and Molecules'' Parr and Yang state in Appendix A, Formula (A.33) If F ist a functional that depends on a parameter \lambda, that is F[f(x,\lambda)] then: \frac{\partial F}{\partial \lambda} = \int \frac{\delta F}{\delta f(x)}...
  5. Z

    What is the meaning of the norm of a linear functional?

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  6. B

    Seeking for Functional Analysis problems solution

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  7. R

    Euler lagrangian equation associated with the variation of a given functional

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  8. S

    Dirichlet eta approximate functional equation

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  9. N

    Why some functional integral(in QTF theo)of a product equal product of two the integr

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  10. S

    Bilinear Form & Linear Functional: Symmetric & Coercive?

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  11. Y

    (n-1)-dimensional subspace is the null space of a linear functional

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  12. M

    Functional Relationship of 555 Monostable PWM

    Hi- This lab question has me stumped... Homework Statement Derive an equation describing the effect of the control voltage on the pules width for a 555 pulse width modulation circuit (monostable circuit). Homework Equations width = 1.1*R*C The Attempt at a Solution I took...
  13. A

    Can *YOU* understand this functional analysis proof?

    My professor tried to show the following in lecture the other day: If T is a linear operator on a Hilbert space and (Tz,z) is real for every z in H, then T is bounded and self-adjoint. Below, I use (*,*) to indicate the Hilbert space inner product. He told us to use the identity (which I've...
  14. F

    How do I minimize a function with a constraint using Lagrange-Euler method?

    I am working on a functional and I need to find its minimum, the conventional procedure is to use Lagrange-Euler method and find the minimum state of the function, but if I need to impose a constraint to the function, I don't know what I need to do J=int(F(t, f(t), a, b)) minimize(f) and...
  15. X

    Conway Functional Analysis text example?

    Hello, I'm reading through John Conway's A Course in Functional Analysis and I'm having trouble understanding example 1.5 on page 168 (2nd edition): Let (X, \Omega, \mu) and M_\phi : L^p(\mu) \to L^p(\mu) be as in Example III.2.2 (i.e., sigma-finite measure space and M_\phi f = \phi f is a...
  16. N

    Grassman number in functional quantization?

    Please teach me this: When calculating something with Grassman numbers without changing order of the numbers,then there are nothing different from ordinary numbers.So I think it would be contrary if we define the complex conjugation of a product of two Grassman numbers to reverse the order of...
  17. P

    The probability functional description of a wave

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  18. A

    Equivalent definitions for the norm of a linear functional

    Can someone please explain why the following three definitions for the norm of a bounded linear functional are equivalent? \| f \| = \sup_{0 < \|x\| < 1} \frac{|f(x)|}{\| x \|}, and \| f \| = \sup_{0 < \| x \| \leq 1} \frac{|f(x)|}{\| x \|}, and \| f \| = \sup_{\| x \| = 1}...
  19. J

    Density Functional Theory and the Local Density Approximation

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  20. K

    Differentiating a Linear Functional

    Hey All, Here's a stupid and probably ridiculously easy question, but I want to make sure that I have it right. Let G be a Lie group with Lie algebra \mathfrak g . Assume that H \in \mathfrak g and \phi \in \mathfrak g^* the algebraic dual. Assume that X(t) is an integral curve...
  21. M

    How to Solve Functional Inequality with Multiple Unknowns?

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  22. Fredrik

    Functional analysis, projection operators

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  23. S

    A simple functional derivative

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  24. S

    Removing Functional Groups from a Molecule: Methods and Possibilities

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  25. S

    What is meant by functional derivative?

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  26. P

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  27. S

    Stationary points of functional

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  28. dav2008

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  29. E

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  30. Rasalhague

    Exploring Roger Penrose's Functional Derivative Notation

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  31. S

    Start Learning Density Functional Theory: A Guide for Beginners

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  32. I

    Existence of linear functional

    Let V be a finite-dimensional vector space over the field F and let T be a linear operator on V. Let c be a scalar and suppose there is a non-zero vector \alpha in V such that t \alpha = c \alpha. Prove that there is a non-zero linear functional f on V such that T^{t}f=cf, where T^{t}f=f\circ T...
  33. S

    How Can the Energy Stored in Functional Groups be Determined?

    How would I find the amount of energy that is stored in a particular functional group? I know things like Azide, Nitro, Alkynyl, Cyanides, etc. would all store a lot of energy.
  34. Z

    What is the difference between a function and a functional?

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  35. T

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  36. A

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  37. A

    Proving Functional Derivative for Current Research - Alice

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  38. C

    Functional analysis applications

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  39. B

    Functional iteration and convergence

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  40. C

    An expression in functional analysis

    Are there any theorems concerning this expression \frac{1}{z}f\left(\frac{1}{z}\right). I appreciate posts of any theorems you can think of.
  41. M

    Functional determinants and gauge invariance

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  42. I

    Functional analysis convergence question

    If X is Banach space and F:X \rightarrow X is a linear and bounded map and that F^n(x)\rightarrow0 pointwise .. How can I show that it converges to zero uniformly also? Thanks
  43. P

    How to Derive the Relation for W_{\varepsilon}[J] Using the Exponential Map?

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  44. P

    Functional Derivative: Computing the d'Alembert Solution

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  45. A

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  46. K

    Real / Functional Analysis Video Lectures?

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  47. E

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  48. T

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  49. S

    Solving Functional Equations: Tips & Examples

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  50. W

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