Complex derivatives

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I have to solve an ODE with variation of coefficient technique. It's pretty easy but I have no clue what is the first and second derivative of e^ix and e^-ix.
 
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  • #2
arildno
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If i had been a real number, what would the first and second derivatives have been then?
 
  • #3
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e^ix
first
i*e^ix
second
i^2*e^ix

e^-ix
first
-i*e^-ix
second
i^2*e^-ix

p.s. I've read about the Cauchy-Riemann equation but just not sure how to apply it... should I split the exponential in a sin and a cos?
p.s.s. There are probably rules, like exponential function are always derivable or something but I'm not fallowing any complex variables class right now so any insight is appreciated...
 
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  • #4
arildno
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e^ix
first
i*e^ix
second
i^2*e^ix

e^-ix
first
-i*e^-ix
second
i^2*e^-ix
EXACTLY!
And that is precisely what holds when "i" is a complex/imaginary number as well! :smile:
 
  • #5
Gib Z
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When dealing with these things, forget i is anything, just remember its a constant. Then after the actual differentiation, you can remember what it is.
 
  • #6
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Yeah. If

[tex]\exp(ix),\,\,\,x\in \mathbb{R},[/tex]

(which is what it looks like you have) then it's what the above two said. But if you have

[tex]\exp(iz),\,\,\,z\in \mathbb{Z},[/tex]

you need to be more careful. Let us know if that is indeed what you have.
 
  • #7
mathwonk
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what are you doing in a de course ifm you do not know the derivative of e^z?
 
  • #8
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I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.
 
  • #9
HallsofIvy
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I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.
That is exactly what has been answered in each of these responses. For any constant, a, the derivative of [itex]e^{ax}[/itex] is [itex]ae^{ax}[/itex].

That is a result of the very basic fact that the derivative of [itex]e^x[/itex] is [itex]e^x[/itex] (world's easiest derivative!) and the chain rule.
 
  • #10
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I'm doing the same derivative problem & i was wondering if you could give any tips on how to solve the derivative of e^ix? I would really appreciate it. A good reference website, anything assistance at all.
[tex]\frac{d}{dx}(e^{jx})=je^{jx}[/tex]
[tex]\frac{d^2}{dx^2}(e^{jx})=-e^{jx}[/tex]
 
  • #11
mathwonk
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Actually I myself was once in an ode course when I had forgot the derivative of e^x. My solution was to go get a Schaum's outline series of ode and do a lot of problems and review my $$$ off.
 

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