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Sum

 Method with 2 overloads

• Sum​(AngouriMath.​Entity,​AngouriMath.​Entity,​AngouriMath.​Entity,​AngouriMath.​Entity)

  Summary

  A summation of expr as var runs from
  from to to inclusive.
  

  Parameter "expr"

  The summand. It may mention var.

  Parameter "var"

  The index, which this binds — it is not a free variable of the result. Naming
  i works: it is the imaginary unit everywhere else, but declaring it as the index
  is taken as meaning it, throughout this operator and nowhere outside it — and the
  same holds of every other binder in the language. See Binding.
  

  Parameter "from"

  The first value of the index.

  Parameter "to"

  The last value of the index, inclusive.

  Returns

  The sum written out where the bounds are concrete integers and there are not too many
  terms; a closed form where the body is a polynomial in the index, whatever the bounds;
  and an unevaluated Summationf otherwise — so a body this cannot
  read is carried rather than refused.
  

  Remarks

  An empty range sums to 0, which is stated rather than left to fall out of the
  loop. #248 That convention is why a closed form carries a condition: sum(k, k, 1, n) is
  (n + n^2)/2 only where n >= 0, the range being empty and the sum
  0 below that while the polynomial is not.
  

  Example

  using System;
  using AngouriMath;
  using static AngouriMath.MathS;
              
  Console.WriteLine(Sum("k", "k", 1, 10).Simplify());
  Console.WriteLine(Sum("k", "k", 1, "n"));

  
  Prints
  

  55
  sum(k, k, 1, n)

• Sum​(AngouriMath.​Entity,​AngouriMath.​Entity,​AngouriMath.​Entity)

  Summary

  The sum of expr over the members of over as
  var ranges over it, each member counted once: written
  sum(expr, var in over). A sum over the roots of a polynomial is a sum over its
  set of roots, sum(f(w), w in { w : p(w) = 0 }), with no index and no order.
  

  Parameter "expr"

  The term, as a function of var.

  Parameter "var"

  The name that ranges over the set, bound by the sum.

  Parameter "over"

  The set.

  Returns

  The terms added up where the set is finite and its members are known: a listed set of
  numbers, or roots of a polynomial that are rational, roots of a quadratic or roots of a
  binomial -- those of an irreducible cubic or quartic are kept as the sum, radicals
  reading no more simply than it. Evaluated to a
  number, the roots it cannot write are found numerically, all of them or none. Left as
  written otherwise: a set with a symbol among its members, which may coincide, or a set
  that is not finite.
  

  Remarks

  #1285

  Example

  using System;
  using static AngouriMath.MathS;
              
  Console.WriteLine(Sum("w^2", "w", "{ w : w^3 - 2w + 1 = 0 }").Simplify());

  
  Prints
  

  4