Math 2.0

Special functions Definition

Trigonometric integrals

Builds on Sine and cosineImproper integralEuler–Mascheroni constant

Definition

Let sin⁡\sin and cos⁡\cos be the sine and cosine functions.

  1. The sine integral is the function Si⁡ ⁣:ℝ→ℝ\operatorname{Si} \colon \R \to \R given by Si⁡(x)=∫0xsin⁡tt dt,\operatorname{Si}(x) = \int_0^x \frac{\sin t}{t}\, dt, where the integrand is given the value 11 at t=0t = 0.
  2. The cosine integral is the function on x>0x > 0 given by the improper integral Ci⁡(x)=−∫x∞cos⁡tt dt.\operatorname{Ci}(x) = -\int_x^\infty \frac{\cos t}{t}\, dt.
  3. The hyperbolic sine integral and hyperbolic cosine integral are Shi⁡(x)=∫0xsinh⁡tt dt\operatorname{Shi}(x) = \int_0^x \frac{\sinh t}{t}\, dt for real xx and Chi⁡(x)=γ+ln⁡x+∫0xcosh⁡t−1t dt\operatorname{Chi}(x) = \gamma + \ln x + \int_0^x \frac{\cosh t - 1}{t}\, dt for x>0x > 0, where γ\gamma is the Euler–Mascheroni constant.

Remarks

Equivalently, Ci⁡(x)=γ+ln⁡x+∫0xcos⁡t−1t dt\operatorname{Ci}(x) = \gamma + \ln x + \int_0^x \frac{\cos t - 1}{t}\, dt for x>0x > 0.

Example: Si⁡\operatorname{Si} is odd, Si⁡(0)=0\operatorname{Si}(0) = 0, and lim⁡x→∞Si⁡(x)=π2\lim_{x \to \infty} \operatorname{Si}(x) = \frac{\pi}{2} by the Dirichlet integral. The largest value of Si⁡\operatorname{Si} on [0,∞)[0, \infty) is Si⁡(π)≈1.8519\operatorname{Si}(\pi) \approx 1.8519.

Non-example: ∫0xcos⁡tt dt\int_0^x \frac{\cos t}{t}\, dt is not a valid definition of Ci⁡\operatorname{Ci}; it diverges at 00 because cos⁡tt∼1t\frac{\cos t}{t} \sim \frac{1}{t} there. This is why Ci⁡\operatorname{Ci} is defined from ∞\infty, and it has a logarithmic singularity at 00.

The shifted function si⁡(x)=Si⁡(x)−π2=−∫x∞sin⁡tt dt\operatorname{si}(x) = \operatorname{Si}(x) - \frac{\pi}{2} = -\int_x^\infty \frac{\sin t}{t}\, dt is also used.

Reading path

To understand this page from the ground up, read these 73 pages in order. The foundations come first.

  1. Big O notation definition
  2. Harmonic number definition
  3. Improper integral concept
  4. Limit point of a set definition
  5. Local maximum definition
  6. Partition of an interval definition
  7. Sequence definition
  8. Sine and cosine concept
  9. Strictly monotone function definition
  10. Little o notation definition
  11. Limit of a function definition
  12. Local minimum definition
  13. Mesh of a partition definition
  14. Riemann sum definition
  15. Real sequence definition
  16. Subsequence definition
  17. Asymptotic equivalence definition
  18. Continuity at a point definition
  19. Derivative definition
  20. One-sided limit definition
  21. Riemann integrable function definition
  22. Bounded sequence definition
  23. Geometric sequence definition
  24. Limit of a sequence definition
  25. Monotone sequence definition
  26. Series definition
  27. Continuous function definition
  28. Epsilon-delta definition of continuity statement
  29. Fermat's theorem on stationary points statement
  30. Sum of a finite geometric series statement
  31. Monotone convergence theorem for sequences statement
  32. Convergent sequence definition
  33. Every sequence has a monotone subsequence statement
  34. Squeeze theorem statement
  35. Cauchy product definition
  36. Partial sum definition
  37. Power series definition
  38. Intermediate value theorem statement
  39. Composition of continuous functions is continuous statement
  40. One-sided limits of a monotone function exist statement
  41. Bolzano–Weierstrass theorem statement
  42. The sequence r^n tends to zero for |r| < 1 example
  43. Radius of convergence definition
  44. Convergent series definition
  45. Boundedness theorem statement
  46. Chain rule statement
  47. Continuity of the inverse of a strictly monotone continuous function statement
  48. A series of nonnegative terms converges iff its partial sums are bounded statement
  49. Absolute convergence definition
  50. Sum of a series definition
  51. Extreme value theorem statement
  52. Derivative of an inverse function statement
  53. Comparison test statement
  54. Geometric series example
  55. Cauchy product of absolutely convergent series statement
  56. Rolle's theorem statement
  57. Ratio test statement
  58. Mean value theorem statement
  59. Ratio formula for the radius of convergence statement
  60. Second fundamental theorem of calculus statement
  61. The exponential series has infinite radius of convergence statement
  62. A function with positive derivative is strictly increasing statement
  63. Exponential function definition
  64. Exponential of a sum statement
  65. The exponential function is positive statement
  66. The exponential function is its own derivative statement
  67. The exponential function is strictly increasing statement
  68. Natural logarithm definition
  69. Derivative of the logarithm statement
  70. Integral of 1/x statement
  71. Harmonic numbers grow like log n statement
  72. Euler–Mascheroni constant object
  73. Trigonometric integrals definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Trigonometric_integral
About this page
TagT761KPKPV6
URIm2://T761KPKPV6
Kinddefinition
WritingW0 stub
MSC33E20, 33B10
DefinesTrigonometric integrals
UsesSine and cosine, Improper integral, Euler–Mascheroni constant

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