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198 subsubsection\<open>The concept of continuously differentiable\<close> |
198 subsubsection\<open>The concept of continuously differentiable\<close> |
199 |
199 |
200 text \<open> |
200 text \<open> |
201 John Harrison writes as follows: |
201 John Harrison writes as follows: |
202 |
202 |
203 ``The usual assumption in complex analysis texts is that a path \<gamma> should be piecewise |
203 ``The usual assumption in complex analysis texts is that a path \<open>\<gamma>\<close> should be piecewise |
204 continuously differentiable, which ensures that the path integral exists at least for any continuous |
204 continuously differentiable, which ensures that the path integral exists at least for any continuous |
205 f, since all piecewise continuous functions are integrable. However, our notion of validity is |
205 f, since all piecewise continuous functions are integrable. However, our notion of validity is |
206 weaker, just piecewise differentiability... [namely] continuity plus differentiability except on a |
206 weaker, just piecewise differentiability... [namely] continuity plus differentiability except on a |
207 finite set ... [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to |
207 finite set ... [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to |
208 the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this |
208 the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this |