Math 2.0

Real functions Riemann and related integrals Example

Gaussian integral

Builds on Improper integralComparison test for improper integralsExponential function

Example

The improper integral of e−x2e^{-x^2} over ℝ\mathbb{R} converges, and ∫−∞∞e−x2 dx=π.\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}. Convergence follows from the comparison test with e−∣x∣e^{-|x|} for ∣x∣≥1|x| \ge 1. For the value, let II be the integral. In polar coordinates, I2=∫02π∫0∞e−r2r dr dθ=πI^2 = \int_0^{2\pi} \int_0^\infty e^{-r^2} r \, dr \, d\theta = \pi.

Remarks

The function e−x2e^{-x^2} has no elementary antiderivative. So the second fundamental theorem does not give the value. A change of variables gives ∫−∞∞e−ax2 dx=π/a\int_{-\infty}^{\infty} e^{-a x^2} \, dx = \sqrt{\pi / a} for a>0a > 0. Mathlib states this as integral_gaussian. Abraham de Moivre found this type of integral in 1733. Carl Friedrich Gauss published the exact value in 1809.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit at infinity, definition, no evidence levelLimit at infinitydefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded 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 levelSeriesdefinitionRiemann integral, definition, no evidence levelRiemann integraldefinitionSum 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 theoremstatementPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionDefinite integral, definition, no evidence levelDefinite integraldefinitionImproper integral, definition, no evidence levelImproper integraldefinitionOne-sided limits of a monotone function exist, statement, no evidence levelOne-sided limits of a…statementThe 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 seriesdefinitionMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementSum of a series, definition, no evidence levelSum of a seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionComparison test, statement, no evidence levelComparison teststatementComparison test for improper integrals, statement, no evidence levelComparison test for i…statementGeometric series, example, no evidence levelGeometric seriesexampleRatio 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 functiondefinitionGaussian integral, example, no evidence levelGaussian integralexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Limit point of a set definition
  2. Partition of an interval definition
  3. Sequence definition
  4. Limit of a function definition
  5. Mesh of a partition definition
  6. Riemann sum definition
  7. Real sequence definition
  8. Limit at infinity definition
  9. One-sided limit definition
  10. Riemann integrable function definition
  11. Bounded sequence definition
  12. Geometric sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Series definition
  16. Riemann integral definition
  17. Sum of a finite geometric series statement
  18. Monotone convergence theorem for sequences statement
  19. Convergent sequence definition
  20. Squeeze theorem statement
  21. Partial sum definition
  22. Power series definition
  23. Definite integral definition
  24. Improper integral definition
  25. One-sided limits of a monotone function exist statement
  26. The sequence r^n tends to zero for |r| < 1 example
  27. Radius of convergence definition
  28. Convergent series definition
  29. Monotonicity of the integral statement
  30. A series of nonnegative terms converges iff its partial sums are bounded statement
  31. Sum of a series definition
  32. Absolute convergence definition
  33. Comparison test statement
  34. Comparison test for improper integrals statement
  35. Geometric series example
  36. Ratio test statement
  37. Ratio formula for the radius of convergence statement
  38. The exponential series has infinite radius of convergence statement
  39. Exponential function definition
  40. Gaussian integral example

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Gaussian_integral, lead section
About this page
Tag0DFT1GP1JP
URIm2://0DFT1GP1JP
Aliaseseuler-poisson-integral
Kindexample
WritingW2 usable
MSC26A42, 40A10
Instance ofImproper integral
UsesImproper integral, Comparison test for improper integrals, Exponential function
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