Math 2.0

Functions of a complex variable Theorem

Power series are differentiable inside the radius of convergence

Builds on Power seriesRadius of convergenceTerm-by-term differentiationPower series converge uniformly on smaller discs inside the radius of convergenceCauchy–Hadamard theorem

Statement

Let f(x)=∑n=0∞an(x−c)nf(x) = \sum_{n=0}^{\infty} a_n (x - c)^n be a power series with radius of convergence R>0R > 0. Then ff is differentiable on (c−R,c+R)(c - R, c + R), and f′(x)=∑n=1∞nan(x−c)n−1.f'(x) = \sum_{n=1}^{\infty} n a_n (x - c)^{n-1}. The derived series also has radius of convergence RR.

Remarks

The Cauchy–Hadamard formula gives the radius of the derived series, since nn→1\sqrt[n]{n} \to 1. Then term-by-term differentiation applies on each [c−r,c+r][c - r, c + r] with r<Rr < R. By induction, ff is infinitely differentiable, and an=f(n)(c)/n!a_n = f^{(n)}(c)/n!. So the series is the Taylor series of ff at cc. Mathlib states differentiability as HasFPowerSeriesOnBall.differentiableOn.

Reading path

To understand this page from the ground up, read these 52 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. Bounded sequence definition
  11. Cauchy sequence definition
  12. Geometric sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Series definition
  16. Continuous function definition
  17. Fermat's theorem on stationary points statement
  18. Limit superior definition
  19. Cauchy sequences are bounded statement
  20. Sum of a finite geometric series statement
  21. Monotone convergence theorem for sequences statement
  22. Every sequence has a monotone subsequence statement
  23. Convergent sequence definition
  24. Pointwise convergence definition
  25. Squeeze theorem statement
  26. Partial sum definition
  27. Power series definition
  28. Bolzano–Weierstrass theorem statement
  29. The sequence r^n tends to zero for |r| < 1 example
  30. Radius of convergence definition
  31. Convergent series definition
  32. Uniform convergence definition
  33. Boundedness theorem statement
  34. Cauchy criterion for sequences statement
  35. Uniform convergence on a set definition
  36. Absolute convergence definition
  37. A series of nonnegative terms converges iff its partial sums are bounded statement
  38. Series of functions definition
  39. Sum of a series definition
  40. Extreme value theorem statement
  41. Uniformly Cauchy sequence of functions definition
  42. Geometric series example
  43. Comparison test statement
  44. Rolle's theorem statement
  45. Root test statement
  46. Weierstrass M-test statement
  47. Mean value theorem statement
  48. Cauchy–Hadamard theorem statement
  49. Power series converge uniformly on smaller discs inside the radius of convergence statement
  50. Uniform convergence of derivatives allows term-by-term differentiation statement
  51. Term-by-term differentiation statement
  52. Power series are differentiable inside the radius of convergence statement

Where this leads

27 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Power_series, Differentiation and integration
About this page
Tag002X7EK91H
URIm2://002X7EK91H
Kindstatement, theorem
WritingW0 stub
MSC30B10, 26A24
StatusThis copy of the site has no build data. Run m2 status 002X7EK91H.
UsesPower series, Radius of convergence, Term-by-term differentiation, Power series converge uniformly on smaller discs inside the radius of convergence, Cauchy–Hadamard theorem

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