complex inputs function

ltn

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Joined
Oct 6, 2021
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Hello!
I've encountered problem:
01.png
and best I came up with is:
t = 3 + ih
-ih = -t + 3
ih = t - 3
2 + 4(t-3)
4t - 10
f'(3) = 4???

I suppose, this shouldn't be solved that way, so my question is: how can this type of problem be solved?

Thank you in advance!
 
Is [imath]h[/imath] real? If [imath]f[/imath] is a real valued function, how can [math]f(3+2ih) = 2+ 4ih[/math]?
 
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I searched for the problem to see if it has any context that would explain the idea of a real-valued function (for real inputs?) that can also have complex inputs and outputs; the problem is found here, which appears to be rather specialized (though I couldn't find the explanation I hoped to find):


My best guess is that you are expected to use the limit definition of the derivative, and hope it will be meaningful.
 
Hello!
Turns out this problem could be solved using this equation (link for reference)
1634046855126.png
So, eventually:

[imath]f'(x)=\frac{lm(2+4ih)}{h}= 4h/h = 4[/imath]
 
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