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Math.NET Symbolics

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# Math.NET Symbolics

Math.NET Symbolics is a basic open source computer algebra library for .NET, Silverlight and Mono written entirely in F#.

This project does not aim to become a full computer algebra system. If you need such a system, have a look at Axiom or Maxima instead, or for proprietary commercial solutions Maple, Mathematica or Wolfram Alpha.

You'll find a large set of expression and algebraic operator examples in the Unit Tests (yes, they're actually very readable). A few examples:

• `(3Q + 2)*4/6``10/3`.
• `(a/b/(c*a))*(c*d/a)/d``1/(a*b)`
• `(a+b)/(b+a)**2``1/(a + b)`
• `Algebraic.expand ((a+b)**3)``a^3 + 3*a^2*b + 3*a*b^2 + b^3`
• `Exponential.expand (exp(2*x+y))``exp(x)^2*exp(y)`
• `Exponential.contract (exp(x)*(exp(x) + exp(y)))``exp(2*x) + exp(x + y)`
• `Exponential.simplify (1/(exp(x)*(exp(y)+exp(-x))) - (exp(x+y)-1)/((exp(x+y))**2-1))``0`
• `Trigonometric.expand (sin(2*x))``2*sin(x)*cos(x)`
• `Trigonometric.contract (sin(x)**2*cos(x)**2)``1/8 - (1/8)*cos(4*x)`
• `Trigonometric.simplify ((cos(x)+sin(x))**4 + (cos(x)-sin(x))**4 + cos(4*x) - 3)``0`
• `Polynomial.polynomialDivision x (x**3 - 2*x**2 - 4) (x-3)``(3 + x + x^2, 5)`
• `Polynomial.polynomialExpansion x y (x**5 + 11*x**4 + 51*x**3 + 124*x**2 + 159*x + 86) (x**2 + 4*x + 5)``1 + x + (2 + x)*y + (3 + x)*y^2`
• `Polynomial.gcd x (x**7 - 4*x**5 - x**2 + 4) (x**5 - 4*x**3 - x**2 + 4)``4 - 4*x - x^2 + x^3`
• `Rational.rationalize (1+1/(1+1/x))``(1 + 2*x)/(1 + x)`
• `Rational.simplify x ((x**2-1)/(x+1))``-1 + x`
```let taylor (k:int) symbol x a =
let rec impl n nf acc dxn =
if n = k then acc else
impl (n+1) (nf*(n+1)) (acc + (dxn |> Structure.substitute symbol a)/nf*(symbol-a)**n) (Calculus.differentiate symbol dxn)
impl 0 1 zero x |> Algebraic.expand

taylor 3 x (1/(1-x)) 0Q       → 1 + x + x^2
taylor 3 x (1/x) 1Q           → 3 - 3*x + x^2
taylor 3 x (ln(x)) 1Q         → -3/2 + 2*x - (1/2)*x^2
taylor 4 x (ln(x)) 1Q         → -11/6 + 3*x - (3/2)*x^2 + (1/3)*x^3
taylor 4 x (sin(x)+cos(x)) 0Q → 1 + x - (1/2)*x^2 - (1/6)*x^3```

### Literature

• Computer Algebra and Symbolic Computation - Elementary Algorithms, Joel. S. Cohen
• Computer Algebra and Symbolic Computation - Mathematical Methods, Joel. S. Cohen
• Modern Computer Algebra, Second Edition, Joachim von zur Gathen, Jürgen Gerhard
• Symbolic Integration I - Transcendental Functions, Second Edition, Manuel Bronstein
• Concrete Mathematics, Second Edition, Graham, Knuth, Patashnik
• ... and of course the fundamental theory by Euclid, Newton, Gauss, Fermat and Hilbert.

### Project

Windows (.NET):

Maintained by Christoph Rüegg and part of the Math.NET initiative (see also Math.NET Numerics). It is covered under the terms of the MIT/X11 open source license. See also the license file in the root folder. We accept contributions!

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