f(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?

This question was previously asked in
BPSC Lecturer ME Held on July 2016 (Advt. 35/2014)
View all BPSC Lecturer Papers >
  1. x2 + y2
  2. x2 – y2
  3. \(\frac{1}{2}\left( {{x^2} + {y^2}} \right)\)
  4. \(\frac{1}{2}\left( {{x^2} - {y^2}} \right)\)

Answer (Detailed Solution Below)

Option 4 : \(\frac{1}{2}\left( {{x^2} - {y^2}} \right)\)

Detailed Solution

Download Solution PDF

संकल्पना:

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

\(\smallint du = \smallint \left( { x} \right)dx - \smallint ydy\)

\(u = \frac{1}{2}\left( {{x^2} - {y^2}} \right)\)

Latest BPSC Lecturer Updates

Last updated on May 9, 2025

-> The BPSC Lecturere DV will be held on 28th May 2025.

-> The written exam took place on 20th March 2025.

-> BPSC Lecturer notification was released for 6 vacancies.

-> The BPSC recruits eligible candidates for the post of Lecturer for various disciplines in Government Colleges across the state.

-> Candidates must attempt the BPSC Lecturer EC Mock Tests. Candidates can also download and practice from the BPSC Lecturer previous year papers

Get Free Access Now
Hot Links: all teen patti teen patti master golden india teen patti party