Prove that u=e⁻ˣ{xsin(y)-ycos(y)} is harmonic & hence find v & f(z) such that f(z)=u+iv is analytic.

Complex Analysis Theorem from Analytic function
Statement/Theorem /Prove that :-
Prove that u=e⁻ˣ{xsin(y)-ycos(y)} is harmonic and hence find v such that f(z)=u+iv is analytic.
Solution/ Proof:
u=e⁻ˣ{xsin(y)-ycos(y)}
Now
Φ₁(x,y)=(∂u/∂x)=e⁻ˣ{sin(y)}+(-e⁻ˣ){xsin(y)-ycos(y)}
⇒ Φ₁(x,y)=(∂u/∂x)=e⁻ˣ{sin(y)-xsin(y)-ycos(y)} .................(1)
⇒ Φ₁(z,0)=(∂u/∂x)=e⁻ᶻ(0-0-0)
⇒ Φ₁(z,0)=(∂u/∂x)=0
And
Φ₂(x,y)=(∂u/∂y)=e⁻ˣ{xcos(y)+ysin(y)-cos(y)} .................(2)
⇒ Φ₂(z,0)=(∂u/∂y)=e⁻ᶻ{z×1+Z×0-1}
⇒ Φ₂(z,0)=(∂u/∂y)=e⁻ᶻ(z-1)
Again differentiating Equation (1) w.r.t x
(∂²u/∂x²)=e⁻ˣ{-sin(y)}+{sin(y)-xsin(y)-ycos(y)}(-e⁻ˣ)
⇒(∂²u/∂x²)=e⁻ˣ{-sin(y)-sin(y)+xsin(y)+ycos(y)}
⇒(∂²u/∂x²)=e⁻ˣ{-2sin(y)+xsin(y)+ycos(y)} ..........(3)
Again differentiating Equation (2) w.r.t y
(∂²v/∂y²)=e⁻ˣ{-xsin(y)-ycos(y)+sin(y)+sin(y)}
⇒(∂²v/∂y²)=e⁻ˣ{-xsin(y)-ycos(y)+2sin(y)} ..........(4)
Adding equations (3) and (4) we get,
(∂²u/∂x²)+(∂²v/∂y²)=e⁻ˣ{-2sin(y)+xsin(y)+ycos(y)-xsin(y)-ycos(y)+2sin(y)}
⇒(∂²u/∂x²)+(∂²v/∂y²)=e⁻ˣ{0}
-------------------------‐------------------------------------
⇒(∂²u/∂x²)+(∂²v/∂y²)=0
-------------------------‐------------------------------------
This shows that the given funtion is Harmonic function.
∵ given function is harmonic , Using Milne-Thomson method.....
f(z)=∫Φ₁(z,0)dz-i∫Φ₂(z,0)dz+c ......(5)
∵ we have Φ₁(z,0)=(∂u/∂x)=0
And Φ₂(z,0)=(∂u/∂y)=e⁻ᶻ(z-1)
Putting these values in equation (5) we get
⇒f(z)=∫0dz-i∫e⁻ᶻ(z-1)dz+c
⇒f(z)=-i[(z-1)∫e⁻ᶻdz-∫{(d/dz)(z-1)∫e⁻ᶻdz}dz]+c
⇒f(z)=-i[(z-1)(-e⁻ᶻ)-∫(-e⁻ᶻ)dz]+c
⇒f(z)=-i[(z-1)(-e⁻ᶻ)-e⁻ᶻdz]+c
⇒f(z)=ie⁻ᶻ(z-1+1)+c
⇒f(z)=ie⁻ᶻz+c
and we know that
v = Im [f(z)]+c
⇒v = Im [ie⁻ᶻz]+c
⇒v = Im [ie⁻ˣe⁻ⁱʸ(x+iy)]+c
⇒v = Im [ie⁻ˣe⁻ⁱʸx-e⁻ˣe⁻ⁱʸy]+c
⇒v = Im [ixe⁻ˣ{cos(y)-isin(y)}-ye⁻ˣ{cos(y)-isin(y)}]+c
⇒v = Im [ixe⁻ˣcos(y)+xe⁻ˣsin(y)-ye⁻ˣcos(y)+iye⁻ˣsin(y)]+c
⇒v = Im [xe⁻ˣsin(y)-ye⁻ˣcos(y)+ixe⁻ˣcos(y)+iye⁻ˣsin(y)]+c
⇒v = Im [e⁻ˣ{xsin(y)-ycos(y)}+ie⁻ˣ{xcos(y)+ysin(y)}]+c
-------------------------‐------------------------------------
⇒v=e⁻ˣ{xcos(y)+ysin(y)}+c
-------------------------‐------------------------------------
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Hence Proved...!!!
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[@rohitmalah1202]

Sir there is wrong while finding du/dx means del u/0del x

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