document.write( "Question 1172367: If function f = u + iv is analytic when u = Sin(x).Cosh(y) then what is the value of v? \n" ); document.write( "

Algebra.Com's Answer #797405 by ikleyn(52831) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "If function f = u + iv is analytic when u = Sin(x).Cosh(y) then what is the $\"highlight%28cross%28value%29%29\"$ expression of v?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "If the complex-value function  f = u + iv  is analytic, then the Cauchy-Riemann equations are held\r\n" );
document.write( "\r\n" );
document.write( "     =       (1)\r\n" );
document.write( "\r\n" );
document.write( "and\r\n" );
document.write( "\r\n" );
document.write( "     = -     (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(as the reference, see this Wikipedia article https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations ).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    I am very sorry, but in these equations the symbol \"d\" must be a Greek letter (\"d rounded\"), not a \"d\" Latin,\r\n" );
document.write( "    but in this editor I can not reproduce it, unfortunately.  So, concider this \"d\" as \"d rounded\" in all my \r\n" );
document.write( "    formulas in this post).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From equation (1), after differentiating  , we have\r\n" );
document.write( "\r\n" );
document.write( "     = cos(x)*cosh(y)      (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From equation (2), after differentiating  , we have\r\n" );
document.write( "\r\n" );
document.write( "     = -sin(x)*sinh(y)     (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, we need to find function v  as antiderivative from equations  (3) and (4).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is easy to do:  the function  v(x,y) = cos(x)*sinh(y)  satisfies both equations  (3) and (4).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "THEREFORE,  the answer to the problem's question is  v = cos(x)*sinh(y).\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "You may add an arbitrary constant value to \"v\" to provide \"a general view\" to satisfy your professor :).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );