Math 2.0

Special functions Proof

Proof that the complex exponential is its own derivative

Builds on Complex exponentialDisc of convergenceThe exponential series has infinite radius of convergencePower series are holomorphic in the disc of convergenceComplex differentiability

Proof

We differentiate the exponential series term by term.

  1. By definition, exp⁡z=∑n≥0anzn\exp z = \sum_{n \ge 0} a_n z^n with an=1/n!a_n = 1/n!, for each z∈ℂz \in \C.
  2. The radius of convergence of this series is sup⁡{r≥0:(rn/n!)n≥0 is bounded}\sup \set{r \ge 0 : (r^n/n!)_{n \ge 0} \text{ is bounded}}. This is the formula for the real series ∑xn/n!\sum x^n/n!, whose radius is ∞\infty by the exponential series has infinite radius of convergence. So the disc of convergence is ℂ\C.
  3. By power series are holomorphic in the disc of convergence, exp⁡\exp is complex differentiable at each z∈ℂz \in \C, and exp⁡′(z)=∑n=1∞nanzn−1=∑n=1∞nn!zn−1.\exp'(z) = \sum_{n=1}^{\infty} n a_n z^{n-1} = \sum_{n=1}^{\infty} \frac{n}{n!} z^{n-1}.
  4. For n≥1n \ge 1 we have n/n!=1/(n−1)!n / n! = 1/(n-1)!.
  5. Put m=n−1m = n - 1. The series becomes ∑m=0∞zm/m!\sum_{m=0}^{\infty} z^m / m!, which is exp⁡z\exp z.
  6. So exp⁡′(z)=exp⁡z\exp'(z) = \exp z for each z∈ℂz \in \C.

Idea

Differentiating zn/n!z^n/n! gives zn−1/(n−1)!z^{n-1}/(n-1)!, the previous term of the same series. So term-by-term differentiation only shifts the index. The theorem on power series makes term-by-term differentiation legal.

Reading path

To understand this page from the ground up, read these 189 pages in order. The foundations come first. The path takes the proofs Proof of the Weierstrass M-test, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof that absolutely convergent series converge, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the exponential series has infinite radius of convergence, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of strong induction from ordinary induction, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the comparison test, Proof that limits preserve non-strict inequalities, Proof of the factorial formula for binomial coefficients by induction, Proof of the binomial theorem by induction, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that absolutely convergent series in a Banach space converge, Proof that power series are holomorphic in the disc of convergence, Proof of the ratio formula for the radius of convergence, Proof that a series of nonnegative terms converges iff its partial sums are bounded.

  1. Binomial coefficient definition
  2. Complex numbers object
  3. Construction of the reals from Cauchy sequences concept
  4. Definition by recursion concept
  5. Element definition
  6. Extended real number line object
  7. Factorial definition
  8. Group concept
  9. Integers object
  10. Mathematical induction concept
  11. Natural numbers object
  12. Norm concept
  13. Ordered field definition
  14. Peano axioms concept
  15. Proposition concept
  16. Rational numbers object
  17. Real numbers object
  18. Sequence definition
  19. Set bounded above definition
  20. Vector space definition
  21. Proof of Pascal's rule from the recursive definition proof
  22. Complex exponential definition
  23. Power series in the complex plane definition
  24. Real and imaginary parts definition
  25. Set definition
  26. Ring definition
  27. Ring of integers statement
  28. Successor function definition
  29. Archimedean ordered field definition
  30. Conjunction definition
  31. Disjunction definition
  32. Implication definition
  33. Negation definition
  34. Predicate definition
  35. The reals form a field statement
  36. Real sequence definition
  37. Subsequence definition
  38. Infimum definition
  39. Supremum definition
  40. Norm definition
  41. Pascal's rule statement
  42. Disc of convergence definition
  43. Empty set object
  44. Family of sets definition
  45. Ordered pair definition
  46. Commutative ring definition
  47. Intersection definition
  48. Union definition
  49. Addition of natural numbers definition
  50. Biconditional definition
  51. Proof of the Archimedean property in an Archimedean ordered field proof
  52. The successor function is injective statement
  53. Zero is not a successor statement
  54. The rationals form a field statement
  55. Truth table concept
  56. Modus ponens concept
  57. Existential quantifier definition
  58. Universal quantifier definition
  59. Bounded sequence definition
  60. Cauchy sequence definition
  61. Geometric sequence definition
  62. Limit of a sequence definition
  63. Monotone sequence definition
  64. Series definition
  65. Completeness axiom statement
  66. Proof of the factorial formula for binomial coefficients by induction proof
  67. Cartesian product definition
  68. Proof of the commutativity of addition by induction proof
  69. Union of a family of sets definition
  70. Archimedean property statement
  71. Order on the natural numbers definition
  72. Subset definition
  73. Propositional logic concept
  74. The rationals form an ordered field statement
  75. Mathematical proof concept
  76. Uniqueness quantifier definition
  77. Proof of the sum of a finite geometric series proof
  78. Convergent sequence definition
  79. Pointwise convergence definition
  80. Partial sum definition
  81. Power series definition
  82. Factorial formula for binomial coefficients statement
  83. Binary relation definition
  84. Topological space definition
  85. Order on the reals definition
  86. Commutativity of addition statement
  87. Proof of strong induction from ordinary induction proof
  88. Tautology definition
  89. Uniform convergence definition
  90. Sum of a finite geometric series statement
  91. Convergent series definition
  92. Radius of convergence definition
  93. Antisymmetric relation definition
  94. Reflexive relation definition
  95. Transitive relation definition
  96. Proof of the binomial theorem by induction proof
  97. Function definition
  98. Strong induction statement
  99. Uniform convergence on a set definition
  100. Logical equivalence definition
  101. Sum of a series definition
  102. Series of functions definition
  103. Absolute convergence definition
  104. Binomial theorem statement
  105. Preorder definition
  106. Proof of the well-ordering principle from strong induction proof
  107. Metric space definition
  108. Proof of the double negation law by a truth table proof
  109. Partial order definition
  110. Convergence of a sequence in a metric space definition
  111. Open ball definition
  112. Well-ordering principle statement
  113. Double negation law statement
  114. Diameter of a set definition
  115. Total order definition
  116. Metric topology definition
  117. Proof by contradiction concept
  118. Proof of the diameter characterisation of Cauchy sequences proof
  119. Proof that every sequence has a monotone subsequence, by peaks proof
  120. Open set definition
  121. Epsilon characterisation of Cauchy sequences statement
  122. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  123. Every sequence has a monotone subsequence statement
  124. Equivalent metrics definition
  125. Complete metric space definition
  126. The reals form an ordered field statement
  127. Equivalent norms definition
  128. Absolute value definition
  129. Proof of the approximation property of the supremum proof
  130. Proof of the existence of square roots of nonnegative reals proof
  131. Banach space definition
  132. Proof of the triangle inequality for real numbers proof
  133. Proof of the epsilon-N definition of the limit of a sequence proof
  134. Existence of square roots of nonnegative reals statement
  135. Approximation property of the supremum statement
  136. Triangle inequality statement
  137. Square root definition
  138. Epsilon-N definition of the limit of a sequence statement
  139. Proof that the reals are Archimedean from the completeness axiom proof
  140. Proof of the squeeze theorem for sequences proof
  141. Modulus of a complex number definition
  142. Proof that limits preserve non-strict inequalities proof
  143. Proof that convergent sequences are bounded proof
  144. Proof that convergent sequences are Cauchy proof
  145. The reals are Archimedean statement
  146. Proof that Cauchy sequences are bounded proof
  147. Proof of the monotone convergence theorem for sequences proof
  148. Complex differentiability definition
  149. Proof of the triangle inequality for norms proof
  150. Limits preserve non-strict inequalities statement
  151. Proof of the ratio formula for the radius of convergence proof
  152. Convergent sequences are bounded statement
  153. Squeeze theorem statement
  154. Convergent sequences are Cauchy statement
  155. Cauchy sequences are bounded statement
  156. Monotone convergence theorem for sequences statement
  157. Holomorphic function definition
  158. Triangle inequality for norms statement
  159. Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
  160. Proof that a decreasing sequence bounded below converges proof
  161. The sequence r^n tends to zero for |r| < 1 example
  162. Proof that absolutely convergent series in a Banach space converge proof
  163. Decreasing sequence bounded below converges statement
  164. A series of nonnegative terms converges iff its partial sums are bounded statement
  165. Geometric series example
  166. Absolutely convergent series in a Banach space converge statement
  167. Proof of the comparison test proof
  168. Proof of the ratio test proof
  169. Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
  170. Comparison test statement
  171. Bolzano–Weierstrass theorem statement
  172. Ratio test statement
  173. Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
  174. Proof that power series are holomorphic in the disc of convergence proof
  175. Ratio formula for the radius of convergence statement
  176. Cauchy criterion for sequences statement
  177. Power series are holomorphic in the disc of convergence statement
  178. Uniformly Cauchy sequence of functions definition
  179. Proof of the Cauchy criterion for series proof
  180. Cauchy criterion for series statement
  181. Proof that absolutely convergent series converge proof
  182. Absolutely convergent series converge statement
  183. Proof of the Weierstrass M-test proof
  184. Weierstrass M-test statement
  185. Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
  186. Power series converge uniformly on smaller discs inside the radius of convergence statement
  187. Proof that the exponential series has infinite radius of convergence proof
  188. The exponential series has infinite radius of convergence statement
  189. Proof that the complex exponential is its own derivative proof

Where this leads

11 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Exponential_function, Complex exponential
About this page
Tag1ZZ5QVHKX0
URIm2://1ZZ5QVHKX0
Kindproof
WritingW0 stub
MSC33B10
ProvesDerivative of the complex exponential
UsesComplex exponential, Disc of convergence, The exponential series has infinite radius of convergence, Power series are holomorphic in the disc of convergence, Complex differentiability
Changed4 November 2026

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