What is the analytic function f(z) in terms of z whose real part is e−x(xsin⁡y−ycos⁡y)?

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nick1337 · Accepted Answer

Step 1We are givenu(x,y)=e−x(xsin⁡y−ycos⁡y)Solve for v(x,y), the imaginary part of f , by using the Cauchy-Riemann equationsvy=ux and vx=−uyUsing the first equation, we obtainvy=−e−x(xsin⁡y−ycos⁡y)+e−xsin⁡yIntegrating both sides with respect to y, we obtainv(x,y)=−e−x(−xcos⁡y−(ysin⁡y+cos⁡y))−e−xcos⁡y+ϕ(x)=e−x(xcos⁡y+ysin⁡y)+ϕ(x)for some arbitrary function ϕTo find ϕ, we next use the second Cauchy-Riemann equation vx=−uy:−e−x(xcos⁡y+ysin⁡y)+e−xcos⁡y+ϕ′(x)=−e−x(xcos⁡y−(cos⁡y−ysin⁡y))This simplifies to ϕ′(x)=0, and thus ϕ(x)=C for some constant C.Hence,v(x,y)=e−x(xcos⁡yysin⁡y)+CPutting this all together, we obtainf(z)=u(x,y)+iv(x,y)=e−x(xsin⁡y−ycos⁡y)+i(e−x(xcos⁡y+ysin⁡y)+C)=e−x((x+iy)sin⁡y+i(x+iy)cos⁡y)+iC=i(x+iy)e−x(cos⁡y−isin⁡y)+iC=i(x+iy)e−xe−xe−iy+iC=ize−z+iC

star233 · Accepted Answer

Step 1 u=e−x(xsin⁡y−ycos⁡y) u is imaginary We gex an analityc function Let f is the analitic function. (f=u_{x}+i^{v}x

What is the analytic function f(z) in terms of z

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