Kompleks Düzlemde Verilen Bir Fonksiyonun w Düzleminde Tasviri
Sedat Han
FpS 2, Sayfa [10-27]
Elektronik yayın tarihi: 8 Aralık 2025
Problemler:
Aşağıdaki fonksiyonların w düzleminde tasvirlerini elde edin. \(z=x+iy\), \(i^2=-1\) ve a, b, c reel.
\[\begin{equation}\label{2.1}\tag{2.1}
w=az+b,\hspace{5pt} |z^2|=c^2,
\end{equation}\]
\[\begin{equation}\label{2.2}\tag{2.2}
w=z^2,\hspace{5pt} x=a,
\end{equation}\]
\[\begin{equation}\label{2.3}\tag{2.3}
w=z^2,\hspace{5pt} y=ax+b,
\end{equation}\]
\[\begin{equation}\label{2.4}\tag{2.4}
w=z^3,\hspace{5pt} y=b,
\end{equation}\]
\[\begin{equation}\label{2.5}\tag{2.5}
w=\frac{1}{z},\hspace{5pt} |z|=1,
\end{equation}\]
\[\begin{equation}\label{2.6}\tag{2.6}
w=e^{y+ix},\hspace{5pt} |z|=1,
\end{equation}\]
\[\begin{equation}\label{2.7}\tag{2.7}
w=zIm(z),\hspace{5pt} y=ax+b,
\end{equation}\]
\[\begin{equation}\label{2.8}\tag{2.8}
w=\sqrt{z}Re(z),\hspace{5pt} |z|=c^2,
\end{equation}\]
\[\begin{equation}\label{2.9}\tag{2.9}
z=e^{aw+b},\hspace{5pt} z=ci,
\end{equation}\]
\[\begin{equation}\label{2.10}\tag{2.10}
w=\cos^{-1}z, \hspace{5pt} z=ci,
\end{equation}\]
\[\begin{equation}\label{2.11}\tag{2.11}
w=\sinh z,\hspace{5pt} |z|=c^2.
\end{equation}\]