[Turkmath:6710] Kompleks analize giriş I

[Ali Sinan Sertöz]

Kızım sana yazıyorum, Yılmaz sen oku! :)

Let z=sqrt( i/(2 n Pi) ). Then -1/z^2=2 n Pi i. (Here sqrt(-1)=i, and n 
is any integer).

Then exp(-1/z^2)=1 for all integers n.

Clearly as n goes to infinity, z goes to zero.

Also as z goes to zero along the real line exp(-1/z^2) goes to zero.

So we have the two path test to show that exp(-1/z^2) is not continuous 
at the origin.

Conclusion exp(-1/z^2) is not analytic at the origin.

Now fast forward: This function has an essential singularity at the 
origin. In every punctured neighborhood of the origin it takes every 
value, with at most one exception,  infinitely many times. This is Big 
Picard theorem. In our case the exception value is zero.

Dersimiz bitti. Haftaya quiz yapacağım! :)

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Ali Sinan Sertöz
Bilkent Üniversitesi, Matematik Bölümü, 06800 Ankara
Ofis Tel: (312) - 290 1490
Bölüm Sekreterliği: (312) - 266 4377
Fax: (312) - 290 1797
e-posta: sertoz at bilkent.edu.tr
Anasayfa: sertoz.bilkent.edu.tr <http://sertoz.bilkent.edu.tr/>
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