Math 2.0

statement proposition

Addition formulas for hyperbolic functions

Statement

For all real numbers xx and yy, sinh⁡(x+y)=sinh⁡xcosh⁡y+cosh⁡xsinh⁡y,\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y, cosh⁡(x+y)=cosh⁡xcosh⁡y+sinh⁡xsinh⁡y.\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y.

Remarks

Expand the right sides with the definitions and use ex+y=exeye^{x+y} = e^x e^y. With y=xy = x, they give sinh⁡2x=2sinh⁡xcosh⁡x\sinh 2x = 2 \sinh x \cosh x and cosh⁡2x=cosh⁡2x+sinh⁡2x\cosh 2x = \cosh^2 x + \sinh^2 x. Mathlib states the first formula as Real.sinh_add.

Explanations

L1 human

These formulas look like the trigonometric addition formulas. The only change is the plus sign in the formula for cosh⁡\cosh.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementCauchy product, definition, no evidence levelCauchy productdefinitionPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionThe sequence r^n tends to zero for |r| < 1, example, no evidence levelThe sequence r^n tend…exampleRadius of convergence, definition, no evidence levelRadius of convergencedefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionSum of a series, definition, no evidence levelSum of a seriesdefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleCauchy product of absolutely convergent series, statement, no evidence levelCauchy product of abs…statementRatio test, statement, no evidence levelRatio teststatementRatio formula for the radius of convergence, statement, no evidence levelRatio formula for the…statementThe exponential series has infinite radius of convergence, statement, no evidence levelThe exponential serie…statementExponential function, definition, no evidence levelExponential functiondefinitionHyperbolic cosine, definition, no evidence levelHyperbolic cosinedefinitionHyperbolic sine, definition, no evidence levelHyperbolic sinedefinitionExponential of a sum, statement, no evidence levelExponential of a sumstatementAddition formulas for hyperbolic functions, statement, no evidence levelAddition formulas for…statement
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  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Geometric sequence definition
  5. Limit of a sequence definition
  6. Monotone sequence definition
  7. Series definition
  8. Sum of a finite geometric series statement
  9. Monotone convergence theorem for sequences statement
  10. Convergent sequence definition
  11. Squeeze theorem statement
  12. Cauchy product definition
  13. Partial sum definition
  14. Power series definition
  15. The sequence r^n tends to zero for |r| < 1 example
  16. Radius of convergence definition
  17. Convergent series definition
  18. A series of nonnegative terms converges iff its partial sums are bounded statement
  19. Absolute convergence definition
  20. Sum of a series definition
  21. Comparison test statement
  22. Geometric series example
  23. Cauchy product of absolutely convergent series statement
  24. Ratio test statement
  25. Ratio formula for the radius of convergence statement
  26. The exponential series has infinite radius of convergence statement
  27. Exponential function definition
  28. Hyperbolic cosine definition
  29. Hyperbolic sine definition
  30. Exponential of a sum statement
  31. Addition formulas for hyperbolic functions statement

Sources

  1. Web https://en.wikipedia.org/wiki/Hyperbolic_functions, Sums and differences of arguments

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