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2bornot2b

How do we figure out what is going to be the domain of the log function for complex numbers? It is easy to figure out for inverse trigonometric functions, since we can draw the graph. But how to do the same for log function?

  • one year ago
  • one year ago

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  1. UnkleRhaukus
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    yeah

    • one year ago
  2. 2bornot2b
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    @TuringTest can you help?

    • one year ago
  3. TuringTest
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    I am terrible at complex analysis... I also didn't get that @ you sent me; I better post that in feedback :S

    • one year ago
  4. experimentX
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    looks like all numbers ... excluding zero.

    • one year ago
  5. 2bornot2b
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    Thanks for coming!

    • one year ago
  6. TuringTest
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    very welcome! if only I could help ....

    • one year ago
  7. 2bornot2b
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    @experimentX The function is a multiple valued function, so there must be an interval. According to my book it says the interval is -pi to +pi, my question was, how to find that range. I hope now I have clarified the thing.

    • one year ago
  8. experimentX
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    I was talking about log for complex values. Inverse Trigonometric functions have range pi to -pi, because they are (they are periodic ... period of 2pi) <--- any value in terms of pi can be expressed in terms of -pi and +pi |dw:1335459261795:dw|

    • one year ago
  9. experimentX
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    and of course it must be multivariable function (not a function)

    • one year ago
  10. 2bornot2b
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    Yes, that is exactly what I am searching for. For sin inverse, you can easily see from the picture what is going to be the range for principle value. And I have been taught to figure that out seeing the plot of sin inverse. But here in log z, how do I find the range for principle value. That is my question.

    • one year ago
  11. experimentX
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    in a same way we say sin(pi/2) = sin(2pi + pi/2) = sin(4pi + pi/2) = sin(6pi + pi/2) = 1 arcsin(1) = pi/2, 2pi+pi/2, 4pi + pi/2, ...

    • one year ago
  12. experimentX
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    now let's check for log, the domain is going to be all comples plane except 0 ln(z) = ln(e^(ln|z| + iarg(z)) = ln|z| + i arg(z)

    • one year ago
  13. experimentX
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    as long as |z| != zero, i think we will have all values of complex plane as our domain.

    • one year ago
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