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#1 2007-01-24 03:57:43

luca-deltodesco
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Registered: 2006-05-05
Posts: 1,470

find the fault in 1 = 3


square both sides

mulitply both sides by

take natural logarithm of both sides
divide by i.pi

can someone please identify the fault here, i cant find it


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#2 2007-01-24 04:21:48

Dross
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Registered: 2006-08-24
Posts: 325

Re: find the fault in 1 = 3

luca-deltodesco wrote:

take natural logarithm of both sides

Complex logarithms are not unique - you'd normally take the principle value; that is the value of  ln(z) = ln|z| + i x arg(z)  when arg(z) is taken to be the principle argument of z. In this case, another value has been taken, so the equality gained by taking logarithms does not hold.


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#3 2007-01-24 04:26:36

luca-deltodesco
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Re: find the fault in 1 = 3

i get it now, i do remember when plotting the natural logarithm over the complex plane that it did create, what might resemble a black hole creating a hole in space time tongue


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#4 2007-01-24 09:19:17

Patrick
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Registered: 2006-02-24
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Re: find the fault in 1 = 3

luca - could you maybe post that? I'd like to see it smile


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#5 2007-01-24 09:32:38

luca-deltodesco
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Registered: 2006-05-05
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Re: find the fault in 1 = 3

ill make up a quick app to graph it again, i long ago lost it


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#6 2007-01-24 09:36:35

Patrick
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Re: find the fault in 1 = 3

Thanks in advance smile


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#7 2007-01-24 10:30:59

luca-deltodesco
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Registered: 2006-05-05
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Re: find the fault in 1 = 3

well i started building a scan line renderer to render the graph, then realised i couldnt be bothered, but also that wolfram has a graph of it tongue

http://mathworld.wolfram.com/images/interactive/NaturalLogReImAbs.gif

i refer to the top left one of cause, y = re[ ln(x+zi) ]


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#8 2007-01-24 10:35:31

Patrick
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Re: find the fault in 1 = 3

Neat, thanks for finding the link smile


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#9 2007-01-24 11:24:22

luca-deltodesco
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Re: find the fault in 1 = 3

i decided to do it anyways big_smile

the colour is a function of y, perspective correct interpolation over the triangle to give the smooth colouring, it is estimated with 100 points from x,z = -5 -> 5 with a cap on y from -2 to 4

http://denvish.net/ulf/1169681050_comp.php

smile


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#10 2007-01-24 11:31:59

luca-deltodesco
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Registered: 2006-05-05
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Re: find the fault in 1 = 3

heres another rendering, this time with 16x the number of triangles, you can see a few visual artifacts, but otherwise its much nicer than previous rendering tongue

and no, these are not pre rendered, its procedural

http://denvish.net/ulf/1169681454_comp.php


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#11 2007-01-24 19:20:04

luca-deltodesco
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Registered: 2006-05-05
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Re: find the fault in 1 = 3

this one is a bit cooler in my opinion, obviously the rendering size has dropped down, and the polygon count has also dropped back down to original, but it does run pretty fast considering its now real time rendering rather than a single rendering.

http://denvish.net/ulf/1169709520_comp.php


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