Suppose z = x + iy = r*(e^(-i*theta)) , where i=sqrt(=1) , r =|z| , tan(theta) = y/x
Also, suppose that w = -uz - m*ln(z) = A + Bi where i=sqrt(-1) and ln=log_e
Then |dw/dz| = 0 implies, -(u + (m/z) =0 ==> z =-(m/u)
Since w = -uz - m*ln(z) = A + Bi ,
A + Bi = -uz - mln(z) = -(ur(e^(itheta))) - mln(r*(e^(i*theta)))
==> A+ Bi = -uz -m[ln(r) - ln(e^(-itheta)) ]
==> A+ Bi = -uz -mln(r) + mln(e^(-itheta))
==> A + Bi = -uz - mln(r) -(mtheta)i
Let theta = T
Then,
A + Bi = -ur(cos(T) - isin(T)) -mln(r) - (mT)i
which implies,
B = ursin(T) -mT
==> B = uy - marctan(y/x) , since r*sin(theta) = y and tan(theta) =y/x
But, the solution is given as B = -uy - m*arctan(y/x)
Could anyone point out the mistake in my calculation?
Thank you!
Your first line bothers me. You say
z = x + iy = r*(e^(-iθ))
But
z = x + iy = r*(e^(iθ))
My professor has given so in the note, I guess that might be the reason for this confusion.
Thank you for pointing that out!
To identify the mistake in your calculation, let's go through the steps again and analyze each part carefully:
First, let's rewrite the expression for w:
w = -uz - m*ln(z)
Let's substitute z = r*(e^(-i*theta)):
w = -ur*(e^(-i*theta)) - m*ln(r*(e^(-i*theta)))
Now, let's simplify the expression by expanding the natural logarithm term:
w = -ur*(e^(-i*theta)) - m*(ln(r) + ln(e^(-i*theta)))
Recall that ln(e^x) = x, so we can simplify further:
w = -ur*(e^(-i*theta)) - m*(ln(r) -i*theta)
Now, let's separate the real and imaginary parts of w.
Real part (A):
A = -ur*cos(theta) - m*ln(r)
Imaginary part (B):
B = -ur*sin(theta) + m*theta
Using the relationship tan(theta) = y/x, we can rewrite sin(theta) and theta:
sin(theta) = y/r
theta = arctan(y/x)
Substituting these values into the expression for B:
B = -ur*(y/r) + m*arctan(y/x)
B = -uy + m*arctan(y/x)
So, the correct expression for B is actually B = -uy + m*arctan(y/x), not B = -uy - m*arctan(y/x) as you mentioned.
Therefore, there seems to be an error in the sign of the second term in your calculation.