## anonymous one year ago Re f=u = -y sinh y cos x + x sin x cosh y, find f ? (Re: real part of f)

1. anonymous

if i remember correctly you have $$u_x=v_y$$ so you could try differentiating and then integrate, but the derivative is such a big mess i think maybe there is some other method

2. anonymous

but f = u(x,y) + i v(x,y)

3. anonymous

i don't know the imaginary part of f

4. anonymous

ux=(ysinh(y)+cosh(y))sin(x)+cosh(y)xcos(x)

5. anonymous

v=integral of vy= sin(x)(sinh(y)+(y−1)ey−(−y−1)e−y2)+xcos(x)sinh(y)+C

6. anonymous

and then f= u+iv is that correct ?

7. Loser66

@oldrin.bataku

8. anonymous

so we want the harmonic conjugate of $$u$$ $$u(x,y) = -y\sinh y\cos x + x\sin x\cosh y$$ if $$f(x+iy)=u(x,y)+iv(x,y)$$ is holomorphic we satisfy the Cauchy-Riemann equations: $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$ so we want a $$v$$ satisfying $$\frac{\partial v}{\partial y}=\frac{\partial u}{\partial x}=y\sinh y\sin x+\sin x\cosh y+x\cos x\cosh y\\\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}=\sinh y\cos x+y\cosh y\cos x-x\sin x\sinh y$$

9. anonymous

so: $$v(x,y)=\int(y\sinh y\sin x+\sin x\cosh y+x\cos x\cosh y)\, dy\\\qquad=\sin x\int y\sinh y\, dy+(\sin x+x\cos x)\sinh y\\\qquad=(y\cosh y-\sinh y)\sin x+(\sin x+x\cos x)\sinh y+f(x)\\\qquad=y\cosh y\sin x+x\sinh y\cos x+f(x)\\v(x,y)=\int(\sinh y \cos x+y\cosh y\cos x-x\sin x\sinh y)dx\\\qquad=\sinh y\sin x+y\cosh y\sin x-\sinh y\int x\sin x\,dx\\\qquad=(\sinh y+y\cosh y)\sin x+(x\cos x-\sin x)\sinh y+g(y)\\\qquad=y\cosh y\sin x+x\cos x\sinh y+g(y)$$ so valid $$v$$ are of the form $$v(x,y)=y\cosh y\sin x+x\sinh y\cos x+C$$

10. anonymous

thank you so much