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2bornot2b

  • 2 years ago

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?

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  1. UnkleRhaukus
    • 2 years ago
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    yeah

  2. 2bornot2b
    • 2 years ago
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    @TuringTest can you help?

  3. TuringTest
    • 2 years ago
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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

  4. experimentX
    • 2 years ago
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    looks like all numbers ... excluding zero.

  5. 2bornot2b
    • 2 years ago
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    Thanks for coming!

  6. TuringTest
    • 2 years ago
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    very welcome! if only I could help ....

  7. 2bornot2b
    • 2 years ago
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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.

  8. experimentX
    • 2 years ago
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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|

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

  10. 2bornot2b
    • 2 years ago
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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.

  11. experimentX
    • 2 years ago
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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, ...

  12. experimentX
    • 2 years ago
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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)

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

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