Expert answers to "Determine whether the complex functions given below satisfy..."

Globokim8

Answered question

2021-02-17

Determine whether the complex functions given below satisfy the Cauchy-Riemann equations (have derivatives). (?

\in C

)
f(z)=z lm (z)

Arnold Odonnell

Skilled2021-02-19Added 109 answers

Step 1
Given that f(z)= z lm(z)
Let z=x+iy
Here x and y are the real and imaginary part of complex number z.
Now f(z) become,
f(z)=(x+iy)y

= x y + i y^{2}

=u+iv
Here,
u=xy
And

v = y^{2}

Step 2
Evaluate the value of

\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}

Thus,

\frac{\partial u}{\partial x} = y

\frac{\partial u}{\partial y} = x

\frac{\partial v}{\partial x} = 0

\frac{\partial v}{\partial y} = 2 y

Since

\frac{\partial u}{\partial x} \neq q \frac{\partial v}{\partial y} and \frac{\partial u}{\partial y} \neq q - \frac{\partial v}{\partial x}

hence given complex functions f(z) does not satisfy the Cauchy-Riemann equations.

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