The value of the transfer function W (p).

Material z Wikipedia

$W$- Lambert's function The start of the ringing function is up to $f\; (w)\; =\; w\; e\; ^\; w$ for complex $w$... Sign $W\; (x)$ abo $\backslash \; operatorname\; (LambertW)\; (x)$... For any complex $z$ here are the functions of the family:

$z\; =\; W\; (z)\; e\; ^\; (W\; (z))$

$W$-function Lambert can be used in elementary functions. Vona stays in a combination, for example, when the number of trees is increased, as well as when the number of trees is violated.

History

The function of Leonard Eiler's robotics was introduced in 1779 to a rock, even a small independent meaning, and was called until the 1980s. The self-contained function of the boolean was introduced in the system of computer algebra Maple LambertW... The name of Johann Heinrich Lambert was reversed, the oskilia Euler sent his robots to the praise of Lambert, and the oskilka "name one function to the name of Euler boolean".

Bagato value

Functions $f\; (w)$ not іn'єactive on the interval $(-\; \backslash \; infty,\; 0)$, $W\; (z)$є a rich-valued function on $[-\; \backslash \; frac\; (1)\; (e),\; 0)$... Iakshto be intertwined with speech $z\; =\; x\; \backslash \; geqslant-1\; /\; e$ i vimagati $w\; \backslash \; geqslant\; -1$, will be assigned an unambiguous function $W\_0\; (x)$.

Asymptotics

It is important to the nobility of the asymptotic function for pragmatic to some key points. For example, in order to accelerate the increase in efficiency, an hour is given to vikonannya recurrent projects.

$\backslash \; left.W\; (z)\; \backslash \; right\; |\; \_\; (z\; \backslash \; to\; \backslash \; infty)\; =\; \backslash \; log\; (z)\; -\; \backslash \; log\; (\backslash \; log\; (z))$

$\backslash \; left.W\; (z)\; \backslash \; right\; |\; \_\; (z\; \backslash \; to\; -\; \backslash \; frac\; (1)\; (e))\; =\; \backslash \; sqrt\; (2\; (ez\; +\; 1))\; -1$

Інші formulas

$\backslash \; int\_\; (0)\; ^\; (\backslash \; pi)\; W\; \backslash \; bigl\; (2\; \backslash \; cot\; ^\; 2\; (x)\; \backslash \; bigr)\; \backslash \; sec\; ^\; 2\; (x)\; \backslash ;\; \backslash \; mathrm\; dx\; =\; 4\; \backslash \; sqrt\; (\backslash \; pi)$ $\backslash \; int\_\; (0)\; ^\; (+\; \backslash \; infty)\; W\; \backslash \; left\; (\backslash \; frac\; (1)\; (x\; ^\; 2)\; \backslash \; right)\; \backslash ;\; \backslash \; mathrm\; dx\; =\; \backslash \; sqrt\; (2\; \backslash \; pi)$ $\backslash \; int\_\; (0)\; ^\; (+\; \backslash \; infty)\; \backslash \; frac\; (W\; (x))\; (x\; \backslash \; sqrt\; (x))\; \backslash \; mathrm\; dx\; =\; 2\; \backslash \; sqrt\; (2\; \backslash \; pi)$

Power

For additional differentiation of the implicit function, it is possible to deny $z\; \backslash \; ne\; -\; \backslash \; tfrac\; (1)\; (e)$ the function of Lambert is happy with the offensive differential competition:

$(dW\; \backslash \; over\; dz)\; =\; \backslash \; frac\; (1)\; (z)\; \backslash \; frac\; (W\; (z))\; (W\; (z)\; +1).$ $e\; ^\; (-\; c\; x)\; =\; a\_o\; (x-r\_1)\; (x-r\_2)\; ~~\; \backslash \; qquad\; \backslash \; qquad\; (2)$ i de constant r 1 i r 2 roots of a square-law bag. In the case of the decision of the family є function with an argument x, a r i that a o є parameters of the central function. From the point of view, unaffected by those that are given to the stagnation of Lambert's W-function with a hypergeometric function and the “Meijer G” function, it should be similar to the other type of functions. r 1 = r 2 then offending parties to the rivnyannya (2) can be forgiven to the rivnyannya (1), and in such a rank home solution Say goodbye to the standard W-function. Rіvnyannya (2) pokazuє viznachalnі vіdnosini in scalar polі dilatonnoї, s chogo viplivaє virіshennya zadachі vimіryuvannya lіnіynoї gravіtatsії Man til 1 + 1 vimіrah (vimіr expanse that vimіryuvannya hour) at razі nerіvnih weight and takozh virіshennya zadachі dvovimіrnogo statsіonarnogo rіvnyannya Shredіngera of potentsіalom viglyadі delta-functions of Dirac for extra charges in one vimir. $e\; ^\; (-\; cx)\; =\; a\_o\; \backslash \; frac\; (\backslash \; displaystyle\; \backslash \; prod\_\; (i\; =\; 1)\; ^\; (\backslash \; infty)\; (x-r\_i))\; (\backslash \; displaystyle\; \backslash \; prod\_\; (i\; =\; 1)\; ^\; (\backslash \; infty)\; (x-s\_i\; ))\; \backslash \; qquad\; \backslash \; qquad\; \backslash \; qquad\; (3)$ de r i that s i is constant, and xє the function between the internal energy and the center of the core of R. Differential Equivalents from memos.

The stagnation of Lambert's W-Functions in the main problems of physics is not surrounded by standard equivalents (1), as it has recently been shown in the fields of atomic, molecular and optical physics.

Assessed

$W$-function can be roughly calculated behind the addition of a recurrent relationship:

w_ (j + 1) = w_j- \ frac (w_j e ^ (w_j) -z) (e ^ (w_j) (w_j + 1) - \ frac ((w_j + 2) (w_je ^ (w_j) -z) ) (2w_j + 2))

Application programs in movi Python:

import math def lambertW (x, prec = 1e-12): w = 0 for i in xrange (100): wTimesExpW = w * math.exp (w) wPlusOneTimesExpW = (w + 1) * math.exp (w) w - = (wTimesExpW-x) / (wPlusOneTimesExpW- (w + 2) * (wTimesExpW-x) / (2 * w + 2)) if (prec> abs ((x-wTimesExpW) / wPlusOneTimesExpW)): break if ( prec<= abs((x-wTimesExpW)/wPlusOneTimesExpW)): raise Exception, "W(x) не сходится достаточно быстро при x=%f" % x return w

For a close calculation, you can vikoristovuvati formula: !!! The function is similar, but more low by 10% is similar to the function of Lambert

W (x) \ approx \ left \ (\ begin (matrix) 0 (,) 665 \ cdot (1 + 0 (,) 0195 \ ln (x + 1)) \ ln (x + 1) + 0 (,) 04 & \: \ & 0 500 \ end (matrix) \ right.

Write an overview about the article "Lambert W-function"

Posilannya

1. Corless ta ін. (1996). "". Adv. Computational Maths. 5 : 329-359.
2. T. C. Scott, R. B. Mann (2006). "". AAECC (Applicable Algebra in Engineering, Communication and Computing) 17 (1): 41-47. DOI: 10.1007 / s00200-006-0196-1.
3. T. C. Scott, G. Fee, J. Grotendorst (2013). "". SIGSAM (ACM Special Interest Group in Symbolic and Algebraic Manipulation) 47 (185): 75–83.
4. T. C. Scott, G. Fee, J. Grotendorst, W.Z. Zhang (2014). "". SIGSAM 48 (1/2): 42–56.
5. PS Farrugia, RB Mann, TC Scott (2007). "". Class. Quantum Grav. 24 (18): 4647-4659. DOI: 10.1088 / 0264-9381 / 24/18/006.
6. T. C. Scott, M. Aubert-Frécon, J. Grotendorst (2006). "". Chem. Phys. 324 : 323-338. DOI: 10.1016 / j.chemphys.2005.10.031.
7. Maignan, Aude (2016). "Fleshing out the Generalized Lambert W Function". SIGSAM 50 (2): 45-60. DOI: 10.1145 / 2992274.2992275.
8. T. C. Scott, A. Lüchow, D. Bressanini, J. D. Morgan III (2007). "The Nodal Surfaces of Helium Atom Eigenfunctions". Phys. Rev. A 75 : 060101. DOI: 10.1103 / PhysRevA.75.060101.
9. at the package

Urivok characterizing the Lambert W-function

- And the other one is Austrian, with him, bouw, at the beginning of vimazaniy. Yak flour, bili. I am tea, how to clean amunits!
- Scho, Fedeshaw! ... having said win, chi, if the countrymen rest, standing nearer? They all said that Bunapart himself stood at Brunov's.
- Bunapart is worth it! bach, a breach, you fool! I don’t know what! Now a Prusak revoltє. Avstriyak yogo, that means, calm down. Yak win to be reconciled, todi and with Bunapart be sure to be reconciled. And then, it seems, the Brunov Bunaparti is worth it! You can see that you are a fool. You hear more.
- Bach chorti kvartiry'єri! P'yata company, be amazed, is already wrapped in the village, stink of porridge, and it’s not going to go until the moment.
- Give me a crouton to those devil.
- And then giving the tyutyun? That, brother. Well, on, God because of you.
- We would like to have a halt, otherwise it will not be more than five miles away.
- That was any good, as we were given carriages. Їdesh, know: it is important!
- And here, brother, the people are calling the mischievous pishov. There, everything is nachebto Pole, all Russian crowns; but nina, brother, is a sutsilny nimets of pishov.
- Go ahead! - you will feel the cry of the captain.
I in front of the swarm of lavas vibiglo about twenty. The drummer fell asleep, turning around to the faces of the children, and, waving his hand, tightening the soldier's song, he repaired: “I’m not dawning, I’ve taken care of myself ...” and ended with the words: “Those, brothers, glory to us ... was stored at Turechchina and now slept in Austria, only with this snake, but on the place “Kamensky Batko” they inserted the words: “Kutuzov Batko”.
The drummer, the dry and garnished soldier of the rocky magpie, suddenly looked round at the soldiers, having looked at the soldiers with his hands, waving his hands, and threw himself to the ground. Then, having crossed over, all eyes are straightened to a new one, I can gently touch my hands with my hands, I feel invisible, expensive riches over my head, feeling for a few seconds and a rapt with a heart-to-heart.
Oh, vi, my blue, blue!
"Seni new my ...", they heard twenty voices, and a spoonful, unaffected by the severity of ammunition, chewed up ahead and backwards in front of the mouth, twisting shoulders and threatening combs with spoons. The soldiers, swinging their hands in time to the beat of the song, walked openly in a crocus, mimovoly, stroking in the leg. Behind the roti, the sounds of kolis, the grumbling of the resor and the dullness of the horses, were heard.
Kutuzov from the mail turned around to the place. The head commander giving a sign, that people were going to cheer up and cheerfully, and on their guise and on all the guises of their mail, they were pleased with the sounds of a song, punching a dancing soldier and soldiers in a roti, very merrily and happily gone. In the other row, on the right flank, the roti beat out the carriage, mimically falling into each other's blaky-eyed soldier, Dolokhov, who was especially lively and gracefully at the beat of the painter and marveled at the faces of those who were living in such a way. at a time for an hour in a row. Hussar cornet from Kutuzov's mail, which imitating the regimental commander, drove out of the carriage and drive to Dolokhov.
Hussar cornet Zherkov at one hour at Petersburg, lying before that exuberant suspension, like a keruvav Dolokhov. Behind the cordon, Zherkov assisted Dolokhov as a soldier, altogether without respecting his needs. Now, I went to talk about Kutuzov with a petition, and because of the joy of an old friend, I turned to him:
- Friend of the heart, ty yak? - having said wine at the sound of a song, the croc of your horse with a croc of roti.
- Am I a yak? - I feel cold Dolokhov, - yak bachish.
Zhvava's song gave a special meaningful tone of rosy gaiety, as Zherkov spoke, and the immense coldness of Dolokhov's ideas.
- Well, yak okay from the bosses? - By feeding Zherkov.
- Nothing, kind people. Ty yak at the headquarters?
- Commands, damn it.
The stench pushed around.
"I let the falcon go from the right sleeve," said the song, mimovoly zbujyuchi badiore, feeling more cheerful. Rozmova їkh, mabut, bula b іnshoyu, yakby stench did not speak at the sound of songs.
- Is it true, the Austrians were beaten? - Having powered Dolokhov.
- And the devil knows, it seems.
“I’m radium,” Dolokhov said briefly and clearly, as if it were a song.
- Well, come before us inject injuries, Pharaoh will lay it down, - said Zherkov.
- Have you got a lot of pennies?
- Come.
- Not possible. Zarok giving. I don’t smoke, I don’t play, the docks don’t crack.
- So well, right before the first ...
- It will be visible there.
I know the stench has pushed around.
“You come in, if you need it, you can help everything at the headquarters…” Zherkov said.
Dolokhov chuckled.
- Don't worry too much. I’m asking for something, I won’t ask, I myself will.
- Well, I’m so ...
- Well, I am.
- Goodbye.
- Be healthy…
... і high, і far,
To the hometown ...
Zherkov jabbing with his spurs, like three times, hot, interrupting with his feet, I don’t know how to get it, and galloping around, catching the company and calling the carriage, maybe at the beat of the song.

Turning to look around, Kutuzov, a subordinate Austrian general, went to his office and, having clicked the ad'yutant, having ordered the taxes to be paid by the deads of the papary, they lay down until they stood, and the leaves were taken away from the guardian of the Armed Forces. Prince Andriy Bolkonsky from the necessary paperwork leads to the office of the head commander. Before spreading the plan on the table, Kutuzov and the Austrian member of the State Committee for Civil Affairs sat.
- And ... - having said Kutuzov, looking around at Bolkonsky, in a word, ask for the ad'utant of pochekati, і prodovzhuvav French rose.
- I only say one thing, the general, - Kutuzov said from the admitting vitality of the vislovs and іntonatsii, so the word rumbled aloud in the skin without fluff. It can be seen that Kutuzov himself was pleased to hear from himself. - I only seem to say it alone, General, that as if on the right there was a special bazhan, then the will of the greatness of Emperor Franz has long been a viconan. I have long ago become the Duke of the Duke. First of all, my honor, so for me to give more special attention to the leadership of the army, instead of the well-known ministerial general, like that of the great auspices of Austria, and to put all this heavy awareness for me, I would be especially happy. Ale, furnish the strongest for us, general.
I Kutuzov, laughing at such a viraz, didn’t know how: “You may not be able to see me again, but you can absolutely navigate me, if you do not believe me. I, in all, is all on the right. "
The Austrian general didn’t agree with his viglyad, but it wasn’t in the wrong way to tell Kutuzov.
- Nawpaki, - having said in a gruff and angry tone, which so overwhelmed the acceptable meaning of words, how to be immobilized, - Navpaki, the fate of your excellency in the out-of-town right is highly valued by its greatness; ale mi vvazhaєmo, who can help you to relieve the glorious Russian vіyska that head commander is quiet laurels, how the stench sounded to reap in battles, - having finished the phrase.
Kutuzov leaned in, not laughing.
- And I’m so overtaking, rushing on the rest of the leaves, like I’ve added the Archduke Ferdinand’s highness to me, let’s admit that the Austrian vіyska, under the leadership of such a master assistant, like General Mac, now I’m too much troubled Kutuzov.
The general frowned. I would like not to hear positive sounds about the assault of the Austrians, a little more than a lot of furnishings, but they were sensitive; And to that Kutuzov's introduction about the Austrian success is similar to a glamor. Ale Kutuzov laughed lazily, all with the same viraz, as if saying that he could afford it. Great, last-minute leaf, taking him out of the Poppy's army, reading him about the change and about the new strategic camp of the army.

19.2.1. Viznachennya The function of a complex change is not at all related to the outward form of functional depletion. Nagadamo , scho region on the area it is called as if you can see the soundless points of the whole area. Region one-link, as if it were a sub-area, surrounded by an uninterrupted, closed, self-non-intersecting curve, so that there is a whole area around it.

Two areas of complex numbers are visible: C = {z | z = x + iy ) that W = {w | w = u + iv ). Go near the square Z given area D і a rule is set to set the type of skin point
not complex number
... It seems to me that it’s in the region D designated unambiguous function w = f (z ) (anyway visualization
). Region D be called the area of ​​the function, without the function, the functionless meaning of the function (or the image of the area D when presented f .

For skin
put at the type of value
(tobto speck z little image), then the function w = f (z ) be called meaningful.

Functions w = f (z ) be called about long-leaved in the area
if there is a uniquely displayed area D per region
(tobto skin point
maє є one image
, і back, dermal point
maє єdinny prototype
.

19.2.2. The design is clear and the part of the function of complex change... So yak

w = u + iv , z = x + iy , then fallowness w = f (z ) can be written at the viglyadі

w = u + iv = f (z ) = f (x + iy ) = Re f (x + iy ) + i Im f (x + iy ). In such a rank, development of complex-valued fu promotions w = f (z ) complex winterz more efficient use of two functional functionsu = u (x , y ) = Re f (z ), v = v (x , y ) = Im f (z ) two effective winter X , at .

put on: 1. w = z 3. Vistulamo z 3 through X ,at : z 3 = (x + iy ) 3 = x 3 + 3 x 2 i y + 3 x i 2 y 2 + i 3 y 3 =

2. w = e z ... Here

We will formulate many of the powers of the FKP (the function of complex change) in terms of the effective part u (x , y ) that explicit part v (x , y ), so the technique of seeing the tsikh chastin is good enough.

19.2.3. Geometric image of the FKP... Functions w = f (z ) yak bet

u = u (x , y ), v = v (x , y ) to think of the image of the FKP yak a couple of surfaces u (x , y ), v (x , y ) in the trivial space, in a way that is not handy, for that, it is not allowed to comprehend a couple ( u , v ) is a complex number. Inodi image the surface, name the yaku relєfom functions w = f (z ). Qiu to apply the surface of the line of function Arg f (z ); for the obviousness of the singing information, there is enough information in order to know about the change in the function of polar coordinates. However, the universal way of the image of the FKP is the fact that many people are painted, they appear one to one when viewed, how to look. Most often take coordinate lines (Cartesian and polar coordinates) and know their images.

put on... 1. Line function w = a z + b de - fixed complex numbers, a 1 , b 1 - їх design parts, a 2 , b 2 - їх clear parts.

Imagine the function of the superposition view of two functions: w 1 = az і w = w 1 + b ... Visualization
, from the interpretation of the multiplication of numbers in the trigonometric form, to reduce the argument to the number z on arg a that stretching (squeezing) of the th module in | a | raziv; visualization
to bring to the point of suvu: w 1 per vector: b (b 1 , b 2). In such a rank, the line function w = a z + b stretching (for
) skin vector z at | a | times (for the squeezeє yogo in raziv at | a | <1), поворачивает на угол arg a і zrushuє on the vector b ... As a result, everything is directly transformed into a straight line, cola - into cola.

2. Step function w = z 2. The function is easy to see at the top

C + = {z | y = Im z > 0). At show forms w = z 2 = (|z | e i arg z ) 2 = |z | 2 e 2 i arg z ... Otzhe, pіvkolo pass at the colo with a vikolotoy point,

promin - at promin. The entire upper area Z + go to the square W with a wicked positive message.

P clearly displayed in Cartesian coordinates. So yak

w = z 2 = (x + iy ) 2 = x 2 - y 2 + 2 ixy , then u (x , y ) = x 2 - y 2 , v (x , y ) = 2 xy ... We know the images of coordinate lines. Straight y = y 0 go to the curve, parametric rivnyannya yako u = x 2 – y 0 2 ,

v = 2 xy 0 (X - Parameter). Viklyuchauchi X , otrimaєmo rivnyannya paraboli
... Promin
go to u = x 0 2 – y 2 ,

v = 2 x 0 y (parameter y > 0). Viklyuchauchi at , otrimaєmo parabolic throat
.

Z v = 2 x 0 y viplyaє, scho v zberigak sign x 0, then the upper head will be x 0> 0, і lower at x 0 <0. Луч x 0 = 0 go to promin u < 0, v = 0.

We look at the functions w = z 2 at the upper floor Z +, unaffected by those that are marked on the whole area Z for this reason, it’s one-leafed in tsіy napіvploshinі. Lower nap_vploshina C - = {z | y = Im z <0} при отображении w = z 2 also cover the entire area W (behind a vinyat of positive pivosi). Yaksho to look at the whole image of the area Z at the same time, then there are two copies of the area W (Two sheets to cover the area).

On a wide range of applications, an algorithm was used to induce images of lines and regions for every hour of displaying w = f (z ). Yaksho w = u (x , y ) + iv (x , y ), those who want to know the image of the line L : F (x , y ) = 0 when displayed, the requirement is from the system
viklyuchiti zmіnnі X і at ; in the result will be ryvnyannya
the image of the line L near the area W ... Know the image of the region D , surrounded by a closed curve L , you need to know the image of the line, as the image is a closed line, you need to go D in the region, surrounded by a central line, for the name of the whole region.

P rimer: hey z 1 = 1 + i , z 2 = 2 + i , z 3 = 1 + 2 i ... Know the image of the tricot man z 1 z 2 z 3 pіd hour of display w = z 2 .

It is known that kudi is driven into the top of the tricycle. w 1 = z 1 2 = (1 + i ) 2 = 1 + 2i - 1 = 2i ;

w 2 = z 2 2 = (2 + i ) 2 = 4 + 4i - 1 = 3 + 4i ;

w 3 = z 3 2 = (1 + 2i ) 2 = 1 + 4i - 4 = -3 + 4i ... Side z 1 z 2 є partly straight at =at 0 = 1. Qia is straightforward to appear, yak mi bachili, at the parabola
... We need a part of the parabola between the dots w 1 i w 2. Dal, side z 1 z 3 є partly straight X =X 0 = 1, which is displayed in a parabola
; we take a dilyanka of a parabolic circle between dots w 1 i w 3. Side z 2 z 3 lie straight X +at = 3; rivnyannya to the image of the direct recognition of the
change X і at :. Dilyanka of a parabolic circle between dots w 2 that w 3 and give an image of the party z 2 z 3. The image of the trikotnik is awake. It is easy to roll over, the area is surrounded by a tricycle tricycle, to cross at the inside of the tricycle tricycle w 1 w 2 w 3 (for everyone it is enough to know, for example, the image of one point of the whole region).

3. Large state-of-the-art function w = z n , de n - a natural number, similar to the function w = z 2. So yak w = z n = (|z | e i arg z ) n = |z | n e i n arg z , then the image is displayed in n once you have to cut with the top at the point z= 0. Be-like two points z 1 i z 2 with the same modules and arguments that can be derived from a multiple (if no stench), go to one point w , tobto. "glued together" when rendered. Otzhe, vіdobrazhennya diversified in the zhodnіy region, it is necessary to take revenge on such points. The butt of the region, de tse imaging univalently - sector
... The whole sector will be transformed into an area, tobto. near the square W with a wicked positive message. Be-yaka area, laid in sectors less than , univalently displayed in W .

19.2.4. Mezha FKP.

Viznachennya. Come on function w = f (z ) is indicated in the punctured near the point z 0 = x 0 + iy 0. Complex number w 0 = u 0 + iv 0 be called a boundary function when
like for be-like -collections
(> 0) points w 0 you know that taka is punctured -colitsa
specks z 0, for all
value f (z ) lay down
... In other words, yaksho z 0 - vlasna point of the area, then for whatever > 0 maє isnuvati also > 0
followed by inconsistency
(similarly, the value for a non-variable point is written
). In such a rank, my -The value of the inter-FKP will increase the value of the inter-function of one action change; to mean the border, yak zvdi:
.

Unfaithfulness
meansє, scho, abo. For the modulus of complex numbers, all the basic powers of the absolute value, zokrem are true, so it is easy to deny that it is

... In such a rank, the function of complex change is equally strong between the two functions. u (x , y ) that v (x , y ) two effective winter. To this, all the theorems about between functions in points (between the sum of the functions too) are automatically transferred to the complex analysis. So you can bring yourself, if you like, then
(For the beginning of a zero line, there is enough
).

19.2.5. The safety of the FKP. Come on function w = f (z ) is marked on the outskirts of the point z 0 = x 0 + iy 0. The function is called uninterrupted at the point z 0 yaksho:

Yak і at a drop of a line, it is possible to show that w = f (z ) will be uninterrupted at points z 0 = x 0 + iy 0 tode and todeless if functions u (x , y ) that v (x , y ) without interruption at points ( x 0 , y 0), so all the basic theorems about the continuity of functions are carried over to the FKP.

Department of Information Systems

Robot course with automation on the topic: "Analysis and synthesis of automatic control systems".

Viconav:

Option 7

Reversed:

Moscow 2008

Entry 4

Rozrakhunkovo-graphic part: 6

1. The value of the transfer function W (p) 6

2.The value of the transfer function W (p) 7

3. The value of the transfer function W (p) 9

4. The value of the transfer function W (p) 10

5. Razrakhunok transient process of the regulated parameter in the ACS 13

6. The value of the indicators of the quality of the regulation and the maximum regulated parameter. 15

7.Value of indicators of quality control 15

8. Pobudova LACHH not changed part of the rostered ACS 15

9. Pobudova bazhanoi LACHH 17

10. Viznachennya LACHH coriguval lanka 19

11.Design transfer function rozimknoyu ACS behind the bazhany LACHH 19

12.Design transfer function coriguval lanka according to LACHH 20

13. Razrakhunok transition process scored ACS 21

14. Viznachennya stock of the speed of the scored ACS for the amplitude that phase. 21

15. The value of the indicators of the quality of the scored ACS control 23

Visnovok 25

List of vikoristanikh dzherel 26
JOIN

Automatic regulation є the most effective principle of automation in partial automation, if the technical part of the automation is needed simple functions control, interconnection with visualization, analysis, control of different physical values ​​and decisions taken by the operator from the installations, programs or other control signals.

Complex automation has come to replace the private sector, if the automation is not only the control functions, but also the victories of the control signals in order to take decisions, go out to the control purposes. In this hour of the Automatic Regulation System (ATS) є the main technically for the shutdown of automated virobnits, workshops, technological processes

The foldability of the casual automatic systems Significantly grown up. Even in the period of private automation, the stench evolved around the systems of automatic regulation, in exchange for the coordination of those connected by people, now the winnickle needs automatic coordination of the automatic control of the automatic control systems. Moreover, at the first stage, the process of regulation is randomly automated, and on the other, the control processes, which may seem to be of a greater backwardness and folding character.

Theoretically, automatic control can be named two characteristic plant:

· At assignments of ACS to know that assessment of transition processes - the price of analysis of ACS;

· Behind the given transition processes and the main indicators of the development of the ACS - the process of mastering the synthesis of the ACS.

Other zavdannya folds through its ambiguity, it is abundant in its origin in the creative health of a designer. That zzvychay zavdannya synthesis of ACS to be put in between. To get involved, the main part of the system is already set, so call it easy. It is necessary to synthesize coriguval lanka, tobto. vibrate їх scheme and parameters. At the same time, it is necessary that, as a result of the correction of the ACS, the necessary supply of stability, the accuracy of control in the modes that have been established, and the quality of control in the dynamic modes are not ensured.

The synthesis of automatic systems is the main and practical addition to the results, taking into account the theory of automatic regulation and control.

Synthesis of the system - the whole vibration structure and storage elements - the physical nature, design and parameters. With the full power of the system and guilt, the consent of the deeds in advance set up vimogs... It is presented both inherently engineering in terms of dimensions, dimensions, variability, and reliability, so that it is specific - to the static and dynamic powers of the system, the quality of regulation.

Zavdannya given course robotsє analysis of a given system and automatic regulation and synthesis with the method of polishing power.

Easy to understand function

Small. 23

de P is a natural number. Pohidna w " = nzn ~ 1іsnu and іdmіnnna from zero at all points z f 0, z f oo. To that, the image, like the function (10.1), is conformable at all points, z = 0 year z = oo. I want to write down the winter zі w at show forms, z = re l w - pe 1b, then (10.1) to make

(we have already looked at the image (10.1) for the P= 2 for butt 5.1). Zvidsey can be seen, what a cola z = r go to the stake | -sh | = r ", kut 0 ip a 2 it / n, with the top on the cob of coordinates, to lie at the z, displayed on cut 0 at the area of ​​sh. Also, the conformity of the image is broken at the point z =0 : kuti in tsy point to grow up when imaged in P development It is not important to show that the image (10.1) is not є conformable at points z = oo(Try it yourself).

Come on specks zі z-2 taki Z2 = P^ 2. Easy vi

kids, scho Z f 22, that Zo= r "e / n with the vertex on the cob of coordinates.

In order to introduce a func- tion that is literal, we need such a value.

Bagato-valued function of complex change called a rule (law), after a complex number z a lot D type of complex numbers w.

All the functions that were seen earlier (except for the functions of Argz) were unambiguous. The Argz є function is very significant:

de argz - smudge the meaning of the argument that before - be-like the number. Give a term function, scho vikoristovuєtsya without any explanations, relying on respect is unambiguous function; The richness of the previously mentioned functions will be negotiated beforehand.

Nekhai function w = f (z) view area D per region E. Zvorotn_y before the function w = f (z) to be called a function z = g (w), marked on the region E, yak skin complex number w € E put complex numbers at the display z € D, taki f (z) = w.

In other words, the function is ringing up to w = f (z),- the whole rule, behind a clear skin point w € E see the prototypes z € D.

Yaksho function і)= f (r) is univalent D, then the vortex function is univalent (and is also univalent) in E yaksho w = f (z) not one-leaf, then the ringing function will be rich in meaning. For example, sound to function w = z nє richly significant function z - y / w: skin value w, vidminnom vid 0 and oo, vidpovida P rooted nth step, which begins with the formula (2.12). The numbers 0 and have one root:> / 0 = 0,> / oo = oo.

Theorem 10.1. Let the function w = f (z) be univalent and apolitic in the domain D, transforming D onto the domain E and f "(z) φ 0. The vortex function z = g (w) is also apolitical in the area of ​​E and

Delivered. Zafiksuєmo to a point z € Dі vіzmemo priіst Az f 0. Todi, through the unisheet function w= / (g) Aw = f (z + Az) - f (z) it is also not expensive. Tom

Functions w = f (z) ana / shtichna, then she is without interruption in point z. Otzhe, Aw-> 0 for Az-> 0; Az-y 0 at Aw-> 0. Zvidsi

well and it is necessary to bring.

Function argument z = g (tv), zvorotn_y w =/ (- g), є zminna w. Oskilki the argument of the function is often expressed through 2, but for the same reason it means the change zі w that write w = g (z). For example, ringing function up to w = z n sign up yak w = yfz.

The presentation of the function w = y / z. Yak bulo is designated as whish, won є as meaningful. Protect it is possible to have a function on a non-folding annex, below a complex area, on a function w = y / z will become one-to-one that without interruption. There are many that are described. Vіzmemo P copies ("sheets") Do, D,..., D n -i complex area, where the growth of positive conditions, and the growth of one over one (in Fig. 24, a indications of vipadoks P= 4). Let's take that edge of the

Small. 24, a

beyond the area to which one goes from the bottom of the exchange OH(tobto ale napivploshini at D glue from the top edge of the area D-2 etc., do not glue the lower edge of the D n -h from the top edge to the size D n -. Now the glued lower edge of the region has become too large. D n -(In Fig. 24, a tse D 3) from the top edge of the area Do- In a trivial space, such gluing is uncomfortable to work without overflowing with already broken gluing of intermediate sheets. Ale mi, we can respect the gluing that is not intersecting with the front ones (so that the points of the whole gluing are taken into account from the points of the other gluing). Otrimana surface is shown in fig. 24, 6 . Vaughn be called Riemannian surface functions w = fz. Above the dermal point of the complex area, from 0 to 0, sorted out P points of a Riemannian surface. Specks X> 0 dіysnoї pіvosі do not become a blame, splinters of all gluing, roasting over it, vvazhayutsya such that do not change. If you do not have two dots, do not lose the power: z = 0 and z = os. All sheets of the Riemannian surface are glued together at the points, spread over the points z= 0 і z= oo.

Apparently now the function w = s / z to induce Riemannian surfaces. Nagadaєmo, shho yaksho z - re, v? then all root n-th step z z and start by formula (2.12):

Small. 24, b

Coot y> the formulas can be vibrated from any interval of up to 27g; we will be able to allow it manually, 0 ^ ip

Krapkam z = re t lie on the arch Do that gluing Do s D n _1, set at the variance of the root value s before= 0; points to lie on the arch D 1 that glue D s Do, - the value of the root z before= 1. Bzagal, points that lie on D * for 1 < before ^ P- 1, і glued D * z D * ._ i, based on the root value from the data before. It will be prompted by the unambiguous function on the Riemann surface.

It is not important to show that the function is uniquely and unambiguously imaging a Riemann surface over the entire complex area. Fair,

~ - * 2ТГ * 27g (& + 1) „-

sheet before will appear in kut

It is shown that the image is uninterrupted. Yaksho point z lie on the arch D * with a rose, then there is no interruption at the point directly from the formula (10.3) with the fix before. For demonstration

without interruptions at the points of gluing, the contour on the surface of the z= 1 complex surface. Slightly go around the contour from the point z, spreading on the upper edge of the leaf By. Oskilki r = 1, cr = 0, before= 0, then w = y / z= 1. When bypassing the first turn, the contour on the arc Do will f 2i G

G-2 T . . 2 T: _ m

і Vz-> cos - + i sin -. Pereyshovy by gluing on arkush P. mi otrimaєmo, for n n

- f + 2 T . f + 2 T

value, l / g = cos- + g sin- (fragments before = one). Zokrem,

at = 0 would be the same root value, to which they were approaching, going to the lower bank of the ale leaf Do. This means that at the points of gluing per s P function sfz will be uninterrupted. Similarly, the continuity of the root and the transition from Dk-i on D * for 1 ^ before ^ P - 1. Nareshty, bypassing the contour on the sheet D „_ 1 and coming close to the lower edge of the rose, we can make it before = 11 - 1, f-e 2 T, і

tobto. those values ​​that were repaired on the upper edge of the leaf P0. In such a rank, function> / r will be uninterrupted at all points of the Riemannian surface. Yak function, ringing to analytical, won є unambiguous analytical function on the surface (except for points z= 0 і z= oo).

Vіzmemo be-yake colo z= g on a complex area, where I hunt a point z = 0. Tse kolo bude will grab also n point z= oo. Bypass the contour on the Roman surface, which is folded from the points that are spread over the cymbol, we pass from one arch of the Riemann surface to the first. Tom points z= 0 і z= oo called points of shading. Zhodna іnsha point is described by the authorities ns maє: how to take a stake with the center at the point z f 0, z f oo, do not take a point 0 in it, then bring points on the Riemann surface P kіl, not tied one by one. Press down on the skin from them, we can’t see beyond the boundaries of one of that very arkush.

Unambiguous analytical in the area D function f (z) be called regular drink rich-valued functions F (z), designated in the same region, as well as f (z) at the skin point of the area D rank with one of the values F (z) at ts_y points.

Bagato-valued function F (z)є unambiguous that is analytic on its Riemann surface (behind a vignette of points of distribution). Therefore, there is a possibility of seeing in the region D Regular pressure means the possibility of growing the area on a small surface, not growing D and do not waste the points of hair removal. Oblap D I am guilty for a lot of times to fit on one arch or to descend along the glue from one arch to the other (yak kilim by descents). For example, ring 1 z F (z) = sfz, n^ 2, shards of the point of the circle.

roztashovani over a positive statement, guilty to spend one hour on the little leaves, but it’s unfortunate. If you develop a circle behind a radius, then we will grow too young. When tsomu roztashuvati D on the Riemann surface it is possible P ways (i, from the same, see in D p new functions y / z). To see a specific point, it is enough to provide a meaningful function in any point of the region D. Tim himself will try a leaf of the Roman surface, for which you will be able to use the point, and that means that it will take care of the whole area. D.

Appendix 10.2. To see a regular gilka f (z) functions w =

2 = e ttp : - -

Decision. Region Dє a complex area with a growing ale at^ 0. From now on, seen regularly D it is possible. For the formula (10.3)

If you see a gilk / (g), you need to know the value of A *. Oskilki / (1) = g, then it is ip= 0, г = 1, we can

viplya stars before= 1. Otzhe, need a head

Zokrem,

We carried out the induction of the Riemannian surface function w == fz, Razrіzajuchi complex area З vdovzh positivnoї pіvosі. Significantly, we do not use principles to vibrate the line of development: an analogous design can be progressed, developed, for example, bridging any exchange, how to go from the cob of coordinates.

Let z = x + iyEC, tody, for the values ​​e z = e x (cos (y) + i ∙ sin (y)).

The function w = e z is assigned according to all Z, it is analogous to C, since

W = u + iv = ex (cos (y) + i ∙ sin (y)) _ (u = ex cos (y), v = ex sin (y)] _ u, vЄC 1 (R 2) Cauchy-Rimann: ∂u / ∂x = ex cos (y), ∂v / ∂y = ex cos (y), ∂u / ∂y = -ex sin (y), ∂x / ∂x = ex sin ( y) y) _ (∂y / ∂x = ∂v / ∂y, ∂u / ∂y = -∂v / ∂x] _ w = ez is an analogous function in C. (ez) "= ∂ (ex ( cos (y) + i ∙ sin (y))) / ∂x = ex (cos (y) + i ∙ sin (y)) = ez.

sz 1, z 2 ЄC e z 1 ∙ e z 2 = e z 1+ z 2, because ez 1 ∙ ez 2 = ex 1 (cos (y 1) + i ∙ sin (y 1)), ex 2 = (cos (y 2) + i ∙ sin (y 2)) = ex 1+ x 2 (cos (y 1 + y 2) + i ∙ sin (y 1 + y 2)) = ez 1+ z 2. When z = x, the interconnected function w = e z goes to the speech line - the function e x.

The function w = ez is periodic with period T = 2πi, ez + 2π i = ez ∙ e 2π i, e 2π i = e 0 (cos (2π) + i ∙ sin (2π)) = 1 _ ez + 2π i = ez , szЄC.

Let ez 1 = ez 2 multiply by e - z 2: e z1-z2 = 1. Number z 1 -z 2 = T 1 + i ∙ T 2 _ e T 1+ i ∙ T 2 = 1 _ e T 1 ( cosT 2 + i ∙ sinT 2) = 1; e T 1 = 1, cosT 2 = 1, sinT 2 = 0 _ T 1 = 0, T 2 = 2πk _z 1 -z 2 = 2πki _ T = 2πi - period. If the domain D does not cover the points z 1 і z 2 such that z 1 -z 2 = 2πki, kЄZ, then the domain D is the domain of single-tasking functions w = e z. Yak D can take divs. rice.

More on the topic 6. Show function w = ez and the main power.

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