Mizar Commentary on Walter Rudin's Principles of Mathematical Analysis/Some Special Functions

Power Series
8.1 Theorem No reference.

Corollary No reference.

8.2 Theorem No reference.

8.3 Theorem No reference, but see DBLSEQ_1/2.

8.4 Theorem No reference.

8.5 Theorem No reference.

The Exponential and Logarithmic Functions
The series expansion of the exponential function is given via SIN_COS:def 5 and TAYLOR_2:16.

8.6 Theorem (a) Continuity is clustered in SIN_COS, differentiability is given by TAYLOR_1:16. (b) TAYLOR_1:16, SIN_COS:65/66 or INTEGRA8:32. (c) Monotonicity implicitly given by <tt>TAYLOR_1:16</tt>, positivity implicitly there too, but also explicitly by <tt>SIN_COS:52/53</tt>. (d) <tt>SIN_COS:23</tt> or <tt>SIN_COS:46</tt>. <tt>SIN_COS:50</tt> or <tt>SIN_COS2:12</tt> for reals only. $$e^0=1$$ is given by <tt>SIN_COS:51</tt> or <tt>SIN_COS2:13</tt>. (e) No direct reference for the limits but see <tt>TAYLOR_1:16</tt>, where they are implicitly given. (f) Implicitly given by <tt>ASYMPT_2:25</tt> (in combination with <tt>ASYMPT_0:15-17</tt> and <tt>ASYMPT_2:def 1</tt>).

$$E(L(y))=y$$ is given by <tt>POWER:def 3</tt> or <tt>TAYLOR_1:14/15</tt>, $$L(E(x))=x$$ is given by <tt>TAYLOR_1:12/13</tt> or <tt>MOEBIUS3:16</tt>. $$L'(y)=\tfrac{1}{y}$$ is given by <tt>FDIFF_4:1</tt> or <tt>TAYLOR_1:18</tt>. The addition formula for logarithms is given by <tt>POWER:53</tt> or <tt>MOEBIUS3:19</tt>. The limits are again only indirectly given by <tt>TAYLOR_1:18</tt>. $$x^\alpha=e^{\alpha\log x}$$ is given by <tt>FDIFF_6:1</tt>. No reference for $$\log x$$ being lower than any positive power of $$x$$.

The Trigonometric Functions
$$C(x)$$ and $$S(x)$$ are defined in <tt>SIN_COS3:def 2</tt> and <tt>SIN_COS3:def 1</tt> respectively. $$E(ix)=C(x)+iS(x)$$ is given by <tt>SIN_COS3:36</tt> (or more general by <tt>SIN_COS3:19</tt>). $$|E(ix)|=1$$ is given by <tt>SIN_COS:27</tt>. No reference for derivates in complex case, but in real case given by <tt>SIN_COS:63/64</tt>. $$\pi$$ in Mizar is defined as the unique $$x\in(0,4)$$ such that $$\tan\tfrac{\pi}{4}=1$$ (<tt>SIN_COS:def 28</tt>). $$\cos(2\pi)=0$$ is given by <tt>SIN_COS:76/77</tt>, that $$\pi$$ is the smallest positive number with that property is given by <tt>SIN_COS:80/81</tt>. The rest of the properties is given by <tt>SIN_COS3:27-33</tt>.

8.7 Theorem (a) No direct reference, only implicitly by <tt>SIN_COS3:27</tt>. (b) <tt>SIN_COS3:35/34</tt> (complex) or <tt>SIN_COS2:11/10</tt> (real). (c) No reference. (d) No reference.

No reference for the circumference of the unit circle, but see <tt>TOPREALB:def 11</tt> for a parametrization of it.

The Algebraic Completeness of the Complex Field
8.8 Theorem is <tt>POLYNOM5:74</tt>.

Fourier Series
8.9 Definition No reference.

8.10 Definition No reference.

8.11 Theorem No reference.

8.12 Theorem No reference.

8.13 Trigonometric series No reference.

8.14 Theorem No reference.

Corollary No reference.

8.15 Theorem No reference.

8.16 Theorem No reference.

The Gamma Function
8.17 Definition No reference.

8.18 Theorem No reference.

8.19 Theorem No reference.

8.20 Theorem No reference.

8.21 Some consequences No reference.

8.22 Stirling's formula No reference.

Exercises
1. No reference. 2. No reference. 3. No reference. 4.(a) No reference. 4.(b) No reference. 4.(c) No reference. 4.(d) No reference, except for $$x=1$$ in <tt>IRRAT_1:22</tt> (see <tt>IRRAT_1:def 4</tt>). 5. No reference. 6. No reference. 7. No reference. 8. No reference. 9. No reference. 10. No reference. 11. No reference. 12. No reference. 13. No reference, but the result is <tt>BASEL_2:32</tt> (see also <tt>BASEL_1:31</tt>). 14. No reference. 15. No reference. 16. No reference. 17. No reference. 18. No reference. 19. No reference. 20. No reference. 21. No reference. 22. No reference. 23. No reference. 24. No reference. 25. No reference. 26. No reference. 27. No reference. 28. No reference. 29. is <tt>BROUWER:15</tt>. 30. No reference. 31. No reference.