Abstract
Recall that a function which is analytic in the whole complex plane is said to be entire or integral. The simplest entire functions which are not polynomial are \(e^z, \sin z\), and \(\cos z,\)
For since the fabric of the universe is most perfect
and the work of a most wise creator nothing at all
takes place in the universe \(\pi \) which some rule of
maxima or minima does not appear
Leonhard Euler (1707–1783)
Many things are not accessible to intuition at all,
the value of \(\int _0^{\infty } e^{-x^2} dx\) for instance
J. F. Littlewood
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Notes
- 1.
Karl Weierstrass (1815–1897), a German mathematician, is well known for his perfect rigor. Weierstrass’s contribution was first published in a paper entitled “Zan theorieder eindeutiogen avalytischen functioner”. Beiliner Abhand Ianagen, 1876, pp. 11–60 [Math. Werke, Vol. II, 1895, pp. 77–124].
- 2.
A milestone in the history leading up to the proof of the Prime Number Theorem is the earlier work of Pafnuty Lvovich Chebyshev showing that \(\pi (x)\) and \(\frac{x}{\log x}\) go to infinity at the similar rate. This fact is, indeed, a very hard-won piece of mathematics! It was proved in 1896 independently by Jacques Hadamard and Charles de la Vallée Poussin.
- 3.
D. Zagier published an article with the title“Newman’s Short Proof of the Prime Number Theorem” in The American Mathematical Monthly, Vol. 104, No. 8 (Oct., 1997), pp. 705–708.
- 4.
See http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/ for the original German version and an English translation.
- 5.
The proof of the Runge’s Theorem that will be given here was obtained by S. Grahiner (Amer. Math. Monthly, 83 (1976), 807–808).
- 6.
Magnas G. Mittag-Leffler (1846–1927) was a Swedish mathematician, a most colorful personality, loved and respected by all. He was greatly influenced by Weierstrass in his approach. His main contribution was in the theory of functions. He also played a great part in inspiring the later research.
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Pathak, H.K. (2019). Entire and Meromorphic Functions. In: Complex Analysis and Applications. Springer, Singapore. https://doi.org/10.1007/978-981-13-9734-9_9
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