en>Rjwilmsi: /* Doppler effect on intensity */ 10.1073/pnas.79.4.1340

2014-12-14T14:42:54Z

Doppler effect on intensity: 10.1073/pnas.79.4.1340

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

2014-01-29T05:24:57Z

References: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 07:24, 29 January 2014
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	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Andrew~~ and ~~my wife doesn~~'~~t like it at~~ all. ~~Invoicing~~ is what ~~I do~~. ~~My spouse~~ and ~~I reside~~ in ~~Kentucky~~. ~~To climb is~~ some ~~thing I really enjoy doing~~.<br><br>~~My web site~~ ... ~~psychic readings online (~~[~~http:~~//~~harvardcu~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~3035~~/~~info~~/ ~~linked webpage~~])		{{For\|the use of the term '''principal value''' in describing improper integrals\|Cauchy principal value}}
			{{Unreferenced\|date=December 2009}}

			In [[complex analysis]], the '''principal values''' of a [[multivalued function]] are the values along one chosen [[branch (mathematics)\|branch]] of that function, so that it is [[Single-valued function\|single-valued]].

			==Motivation==
			Consider the [[complex logarithm]] function log ''z''. It is defined as the [[complex number]] ''w'' such that
			:<math>e^w = z\,\!</math>
			Now, for example, say we wish to find log i. This means we want to solve
			:<math>e^w = i\,\!</math>
			for ''w''. Clearly iπ/2 is a solution. But is it the only solution?

			Of course, there are other solutions, which is evidenced by considering the position of i in the [[complex plane]] and in particular its [[argument (complex analysis)\|argument]] arg ''i''. We can rotate counterclockwise π/2 radians from 1 to reach i initially, but if we rotate further another 2π we reach i again. So, we can conclude that i(π/2 + 2π) is ''also'' a solution for log i. It becomes clear that we can add any multiple of 2πi to our initial solution to obtain all values for log i.

			But this has a consequence that may be surprising in comparison of real valued functions: log i does not have one definite value! For log ''z'', we have
			:<math> \log{z} = \ln{\|z\|} + i\left(\mathrm{arg}\ z \right)
			= \ln{\|z\|} + i\left(\mathrm{Arg}\ z+2\pi k\right) </math>
			for an integer ''k'', where Arg ''z'' is the (principal) argument of ''z'' defined to lie in the interval <math> (-\pi,\ \pi]</math>. Each value of ''k'' determines what is known as a ''[[branch (mathematics)\|branch]]'' (or ''sheet''), a single-valued component of the multiple-valued log function.

			The branch corresponding to ''k''=0 is known as the ''principal branch'', and along this branch, the values the function takes are known as the ''principal values''.

			==General case==
			In general, if ''f''(''z'') is multiple-valued, the principal branch of ''f'' is denoted
			:<math>\mathrm{pv}\ f(z)</math>
			such that for ''z'' in the domain of ''f'', pv ''f''(''z'') is single-valued.

			===Principal values of standard functions===
			Complex valued [[List of mathematical functions\|elementary functions]] can be multiple valued over some domains. The principal value of some of these functions can be obtained by decomposing the function into simpler ones whereby the principal value of the simple functions are straightforward to obtain.

			====Logarithm function====
			We have examined the [[logarithm function]] above, i.e.,
			:<math>\log{z} = \ln{\|z\|} + i\left(\mathrm{arg}\ z\right).</math>
			Now, arg ''z'' is intrinsically multivalued. One often defines the argument of some complex number to be between -π (exclusive) and π (inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch Arg ''z'' (with the leading capital A). Using Arg ''z'' instead of arg ''z'', we obtain the principal value of the logarithm, and we write
			:<math>\mathrm{pv}\ \log{z} = \mathrm{Log}\ z = \ln{\|z\|} + i\left(\mathrm{Arg}\ z\right).</math>

			====Exponential function====
			So far we have only considered the logarithm function. What about [[Exponential function\|exponents]]?

			Consider <math>z ^\alpha\,</math> with <math> \alpha \in \mathbb{C}</math> . One usually defines ''z''<sup>α</sup> to be ''e''<sup>α log ''z''</sup>. Yet ''e''<sup>α log ''z''</sup> is multiple-valued since we are using log as opposed to Log. Using Log we obtain the principal value of ''z''<sup>α</sup>, i.e.,
			: <math>\mathrm{pv}\ z^\alpha = e^{\alpha \mathrm{Log}\ z}.</math>

			====Square root====
			For a [[complex number]] <math>z=r e^{\phi i}\,</math> the principal value of the [[square root]] is :

			:<math>\mathrm{pv} \sqrt{z} = \sqrt{r} \, e^{i \phi / 2}</math>

			with [[Arg (mathematics)\|argument]] <math>-\pi < \phi \le \pi \,</math>

			====Complex argument====
			[[File:Atan2atan.png\|right\|thumb\|comparison of [[Inverse trigonometric functions\|atan]] and [[atan2]] functions]]

			The principal value of [[Arg (mathematics)\|complex number argument]] measured in [[radian]]s can be defined as:
			* values in the range [0, 2π)
			* values in the range (-π, π].

			To compute these values one can use functions :
			* [[atan2]] with principal value in the range (-π, π]
			* [[Inverse trigonometric functions\|atan]] with principal value in the range (-π/2, π/2]

			==See also==
			*[[Principal branch]]
			*[[Branch point]]

			{{DEFAULTSORT:Principal Value}}
			[[Category:Complex analysis]]

en>Xiong: /* Visualization */ rm babble

2012-08-27T21:11:49Z

Visualization: rm babble

New page

I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Invoicing is what I do. My spouse and I reside in Kentucky. To climb is some thing I really enjoy doing.<br><br>My web site ... psychic readings online ([http://harvardcu.com/index.php?do=/profile-3035/info/ linked webpage])

Relativistic Doppler effect - Revision history

en>Rjwilmsi: /* Doppler effect on intensity */ 10.1073/pnas.79.4.1340

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

en>Xiong: /* Visualization */ rm babble