#sketching complex
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midnight salmon
can you draw the line Im(z) - Re(z) = 1?
wary raft
No …
midnight salmon
that's y - x = 1 on a ordinary cartesian plane
wary raft
I might be cooked
Oh
Would that solve my issue?
And then shading the area within those 3 lines
midnight salmon
yep
wary raft
Really quickly, could you please give me some pointers (if you have time) on how I could attempt part b)?
This is where I’m at right now
noble raven
The vertical segment is rather straightforward, varying the imaginary part would making the graph of the exponential draw an arc centred at the origin
The horizontal segment is also similar, should be a radial segment which, when extended, passes through the origin
The toughest part is the remaining logarithmic spiral mapped from the upward sloping segment
wary raft
How do you recognise how to transform the function in this way
Are you take the exponent of the line?
As z—>e^z
noble raven
The function is the transform
To be precise, I am taking the exponential of each point on the line segments
Since the real parts of all points on a vertical line are the same, they are mapped to points of equal magnitude
Similarly, the imaginary parts of all points on a horizontal line are the same, so they are mapped to points of equal angular argument
wary raft
I'm sorry but im not too sure I understand. So the Im(z) and Re(z) stay the same?
But the y-x=1 line is altered by the exponential?
How would I sketch the remaining logarithmic spiral? Would that stem from the y-x=1 line?
sour pythonBOT
ℝafain
midnight salmon
try it out for specific points on the line (-1, pi/2 - 1 + pi/2 i, i, etc.)
wary raft
What do you mean by try it out?
noble raven
Map those specific points