# Complex analysis problem

• Mulz

#### Mulz

Homework Statement
Find a function that solves the complex problem ## f:C \rightarrow C## that solves ##\frac{df}{dx} = 6x + 6iy##
Relevant Equations
##\Delta u = Re(f) = \frac{df^2}{dx^2} + \frac{df^2}{dy^2}##
Mentor note: Edited to fix LaTeX problems
##f:C \rightarrow C## that solves ##\frac{df}{dx} = 6x + 6iy##

## f(x,y) = 3x^2 + 6xyi + C(y) = (3x^2 + C(y)) + i(6xy) ##

## \Delta u = 0 \rightarrow 6 + C''(y) = 0 \rightarrow C(y) =5 \frac{5}{5}##

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Dragounat said:
Homework Statement:: Find a function that solves the complex problem ## f:C \rightarrow C## that solves ##\frac{df}{dx} = 6x + 6iy##
Relevant Equations:: ##\Delta u = Re(f) = \frac{df^2}{dx^2} + \frac{df^2}{dy^2}##

Mentor note: Edited to fix LaTeX problems
##f:C \rightarrow C## that solves ##\frac{df}{dx} = 6x + 6iy##

## f(x,y) = 3x^2 + 6xyi + C(y) = (3x^2 + C(y)) + i(6xy) ##

## \Delta u = 0 \rightarrow 6 + C''(y) = 0 \rightarrow C(y) =5 \frac{5}{5}##
Note: I fixed all of your LaTeX script.

Your solution for f(x, y) looks fine to me, but I get something different for C(y).

If ##f(x, y) = 3x^2 + 6xyi + C(y)##, then
##f_x = 6x + 6yi## and ##f_y = 6xi + C'(y)##

##\Delta u = 0 \Rightarrow 6 + C''(y) = 0 \Rightarrow C''(y) = -6##
Solve the last equation above to find C(y).