Question
Download Solution PDFf(z) = u(x, y) + iv(x, y) सामिश्र चर z = x + iy का एक विश्लेषणात्मक फलन है। यदि v = xy है, तो u(x, y) किसके बराबर है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
यदि f(z) = u(x, y) + iv(x, y) एक विश्लेषणात्मक फलन है, तो कॉची-रीमैन की स्थिति संतुष्ट होगी।
अर्थात् \(\frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial y}}~and~\frac{{\partial u}}{{\partial y}} = - \frac{{\partial v}}{{\partial x}}\)
गणना:
दिया गया है:
v = xy
\(\frac{{\partial v}}{{\partial y}} = x \Rightarrow \frac{{\partial u}}{{\partial x}} = x\)
\( \frac{{\partial v}}{{\partial x}} = y, \frac{{\partial u}}{{\partial y}} = - \frac{{\partial v}}{{\partial x}} = -y\)
यदि u = f(x, y) है।
\(du = \frac{{\partial u}}{{\partial x}}dx + \frac{{\partial u}}{{\partial y}}dy\)
du = xdx - ydy
दोनों पक्षों का समाकलन करने पर
\(\smallint du = \smallint \left( { x} \right)dx - \smallint ydy\)
\(u = \frac{1}{2}\left( {{x^2} - {y^2}} \right)\)
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