Math 2.0

statement theorem

Exponential of a sum

Statement

For all real numbers xx and yy, exp⁡(x+y)=exp⁡(x)exp⁡(y).\exp(x + y) = \exp(x) \exp(y).

Remarks

Multiply the two series by the Cauchy product. The binomial theorem gives ∑k=0nxkk!yn−k(n−k)!=(x+y)nn!\sum_{k=0}^{n} \frac{x^k}{k!} \frac{y^{n-k}}{(n-k)!} = \frac{(x + y)^n}{n!}. In particular, exp⁡(x)exp⁡(−x)=exp⁡(0)=1\exp(x) \exp(-x) = \exp(0) = 1. So exp⁡\exp is a group homomorphism from (R,+)(\mathbb{R}, +) to the multiplicative group (0,∞)(0, \infty). Mathlib states the formula as Real.exp_add.

Explanations

L1 human

The exponential function turns addition into multiplication. This is the rule ex+y=exeye^{x + y} = e^x e^y for powers.

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To read this node, read these 28 nodes in this order. The foundations come first, and the list ends with this node.

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 functiondefinitionExponential of a sum, statement, no evidence levelExponential of a sumstatement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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. Exponential of a sum statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Functional equation

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