AngouriMath

Navigation

Home Quick start Wiki Almanac Research What's new Try Use Angouri.org GitHub Report a bug Donate Contacts

← Back to list of members

Taylor

 Method with 2 overloads

• Taylor​(AngouriMath.​Entity,​System.​Int32,​System.​ValueTuple{AngouriMath.​Entity.​Variable,​AngouriMath.​Entity.​Variable,​AngouriMath.​Entity}[])

  Summary

  Finds the symbolic expression of terms of the Taylor expansion of the given function,
  https://en.wikipedia.org/wiki/Taylor_series
  

  Parameter "expr"

  The function to find the Taylor expansion of
  

  Parameter "degree"

  The degree of the resulting taylor polynomial
  

  Parameter "exprToPolyVars"

  The variable/s to take the series over, the variable the series will be over,
  and the variable values at which the Taylor polynomial will be found
  (e. g. if you want to find the taylor polynomial of Sin("t") around t=1, and want
  n to take that place in the series, then you may want to use ("t","n","1") for this argument)
  

  Returns

  An expression in the polynomial form over the poly variable/s given in exprToPolyVars

  Example

  using AngouriMath;
  using System;
  using System.Linq;
  using static AngouriMath.MathS;
  using static AngouriMath.MathS.Series;
  
  var (x, y) = MathS.Var("x", "y");
  Console.WriteLine(Sin(x));
  Console.WriteLine(Maclaurin(Sin(x), 1, x));
  Console.WriteLine(Maclaurin(Sin(x), 2, x));
  Console.WriteLine(Maclaurin(Sin(x), 3, x));
  Console.WriteLine(Maclaurin(Sin(x), 4, x));
  Console.WriteLine(Maclaurin(Sin(x), 10, x).Simplify());
  Console.WriteLine("----------------------");
  var expr = Sin(x) + Cos(y);
  Console.WriteLine(expr);
  Console.WriteLine(Maclaurin(expr, 6, x, y).Simplify());
  Console.WriteLine("----------------------");
  Console.WriteLine(expr);
  Console.WriteLine(Taylor(expr, 6, (x, 1)));
  Console.WriteLine("----------------------");
  Console.WriteLine(expr);
  Console.WriteLine(Taylor(expr, 6, (x, 1), (y, 5)));
  Console.WriteLine("----------------------");
  Console.WriteLine(expr);
  Console.WriteLine(Taylor(expr, 6, (x, "z_1", 1), (y, "z_2", 5)));
  Console.WriteLine("----------------------");
  var first3Terms = 
      TaylorTerms(expr, (x, x, 0), (y, y, 1))
      .Take(3);
  var first6Terms =
      TaylorTerms(expr, (x, x, 0), (y, y, 1))
      .Take(6);
  foreach (var term in first6Terms)
      Console.WriteLine($"Received {term}");

  
  Prints
  

  sin(x)
  0
  0 + x
  0 + x + 0
  0 + x + 0 + -x ^ 3 / 3!
  x ^ 9 / 362880 - x ^ 7 / 5040 + x ^ 5 / 120 - x ^ 3 / 6 + x
  ----------------------
  sin(x) + cos(y)
  1 + (6 * x ^ 5 - 120 * x ^ 3) / 720 + x + (2 * y ^ 4 - 24 * y ^ 2) / 48
  ----------------------
  sin(x) + cos(y)
  sin(1) + cos(y) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + cos(1) * (-1) * (x - 1) ^ 3 / 3! + -sin(1) * (-1) * (x - 1) ^ 4 / 4! + cos(1) * (-1) * (-1) * (x - 1) ^ 5 / 5!
  ----------------------
  sin(x) + cos(y)
  sin(1) + cos(5) + cos(1) * (x - 1) + -sin(5) * (y - 5) + (-sin(1) * (x - 1) ^ 2 + cos(5) * (-1) * (y - 5) ^ 2) / 2! + (cos(1) * (-1) * (x - 1) ^ 3 + -sin(5) * (-1) * (y - 5) ^ 3) / 3! + (-sin(1) * (-1) * (x - 1) ^ 4 + cos(5) * (-1) * (-1) * (y - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (x - 1) ^ 5 + -sin(5) * (-1) * (-1) * (y - 5) ^ 5) / 5!
  ----------------------
  sin(x) + cos(y)
  sin(1) + cos(5) + cos(1) * (z_1 - 1) + -sin(5) * (z_2 - 5) + (-sin(1) * (z_1 - 1) ^ 2 + cos(5) * (-1) * (z_2 - 5) ^ 2) / 2! + (cos(1) * (-1) * (z_1 - 1) ^ 3 + -sin(5) * (-1) * (z_2 - 5) ^ 3) / 3! + (-sin(1) * (-1) * (z_1 - 1) ^ 4 + cos(5) * (-1) * (-1) * (z_2 - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (z_1 - 1) ^ 5 + -sin(5) * (-1) * (-1) * (z_2 - 5) ^ 5) / 5!
  ----------------------
  Received cos(1)
  Received x + -sin(1) * (y - 1)
  Received cos(1) * (-1) * (y - 1) ^ 2 / 2!
  Received (-x ^ 3 + -sin(1) * (-1) * (y - 1) ^ 3) / 3!
  Received cos(1) * (-1) * (-1) * (y - 1) ^ 4 / 4!
  Received (x ^ 5 + -sin(1) * (-1) * (-1) * (y - 1) ^ 5) / 5!

• Taylor​(AngouriMath.​Entity,​System.​Int32,​System.​ValueTuple{AngouriMath.​Entity.​Variable,​AngouriMath.​Entity}[])

  Summary

  Finds the symbolic expression of terms of the Taylor expansion of the given function,
  Wikipedia

  Parameter "expr"

  The function to find the Taylor expansion of
  

  Parameter "degree"

  The degree of the resulting taylor polynomial
  

  Parameter "exprVariables"

  The variable/s to take the series over (and the variable in the resulting series),
  plus the variable values at which the Taylor polynomial will be found
  (e. g. if you want to find the taylor polynomial of Sin("t") around t=1, then you may want to use ("t","1") for this argument)
  

  Returns

  An expression in the polynomial form over the expression variable/s given in exprVariables

  Example

  using AngouriMath;
  using System;
  using System.Linq;
  using static AngouriMath.MathS;
  using static AngouriMath.MathS.Series;
  
  var (x, y) = MathS.Var("x", "y");
  Console.WriteLine(Sin(x));
  Console.WriteLine(Maclaurin(Sin(x), 1, x));
  Console.WriteLine(Maclaurin(Sin(x), 2, x));
  Console.WriteLine(Maclaurin(Sin(x), 3, x));
  Console.WriteLine(Maclaurin(Sin(x), 4, x));
  Console.WriteLine(Maclaurin(Sin(x), 10, x).Simplify());
  Console.WriteLine("----------------------");
  var expr = Sin(x) + Cos(y);
  Console.WriteLine(expr);
  Console.WriteLine(Maclaurin(expr, 6, x, y).Simplify());
  Console.WriteLine("----------------------");
  Console.WriteLine(expr);
  Console.WriteLine(Taylor(expr, 6, (x, 1)));
  Console.WriteLine("----------------------");
  Console.WriteLine(expr);
  Console.WriteLine(Taylor(expr, 6, (x, 1), (y, 5)));
  Console.WriteLine("----------------------");
  Console.WriteLine(expr);
  Console.WriteLine(Taylor(expr, 6, (x, "z_1", 1), (y, "z_2", 5)));
  Console.WriteLine("----------------------");
  var first3Terms = 
      TaylorTerms(expr, (x, x, 0), (y, y, 1))
      .Take(3);
  var first6Terms =
      TaylorTerms(expr, (x, x, 0), (y, y, 1))
      .Take(6);
  foreach (var term in first6Terms)
      Console.WriteLine($"Received {term}");

  
  Prints
  

  sin(x)
  0
  0 + x
  0 + x + 0
  0 + x + 0 + -x ^ 3 / 3!
  x ^ 9 / 362880 - x ^ 7 / 5040 + x ^ 5 / 120 - x ^ 3 / 6 + x
  ----------------------
  sin(x) + cos(y)
  1 + (6 * x ^ 5 - 120 * x ^ 3) / 720 + x + (2 * y ^ 4 - 24 * y ^ 2) / 48
  ----------------------
  sin(x) + cos(y)
  sin(1) + cos(y) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + cos(1) * (-1) * (x - 1) ^ 3 / 3! + -sin(1) * (-1) * (x - 1) ^ 4 / 4! + cos(1) * (-1) * (-1) * (x - 1) ^ 5 / 5!
  ----------------------
  sin(x) + cos(y)
  sin(1) + cos(5) + cos(1) * (x - 1) + -sin(5) * (y - 5) + (-sin(1) * (x - 1) ^ 2 + cos(5) * (-1) * (y - 5) ^ 2) / 2! + (cos(1) * (-1) * (x - 1) ^ 3 + -sin(5) * (-1) * (y - 5) ^ 3) / 3! + (-sin(1) * (-1) * (x - 1) ^ 4 + cos(5) * (-1) * (-1) * (y - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (x - 1) ^ 5 + -sin(5) * (-1) * (-1) * (y - 5) ^ 5) / 5!
  ----------------------
  sin(x) + cos(y)
  sin(1) + cos(5) + cos(1) * (z_1 - 1) + -sin(5) * (z_2 - 5) + (-sin(1) * (z_1 - 1) ^ 2 + cos(5) * (-1) * (z_2 - 5) ^ 2) / 2! + (cos(1) * (-1) * (z_1 - 1) ^ 3 + -sin(5) * (-1) * (z_2 - 5) ^ 3) / 3! + (-sin(1) * (-1) * (z_1 - 1) ^ 4 + cos(5) * (-1) * (-1) * (z_2 - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (z_1 - 1) ^ 5 + -sin(5) * (-1) * (-1) * (z_2 - 5) ^ 5) / 5!
  ----------------------
  Received cos(1)
  Received x + -sin(1) * (y - 1)
  Received cos(1) * (-1) * (y - 1) ^ 2 / 2!
  Received (-x ^ 3 + -sin(1) * (-1) * (y - 1) ^ 3) / 3!
  Received cos(1) * (-1) * (-1) * (y - 1) ^ 4 / 4!
  Received (x ^ 5 + -sin(1) * (-1) * (-1) * (y - 1) ^ 5) / 5!