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Principal Branch of Log Principal branch of log Continue ←Complex → consider $z any non-grainy number. We would like to solve for $w$, the equation starteqnarraylabel log1 e'w'z. endeqnarray If $Theta'textbf (z)$ with $-'pi zlt; Theta leq pi$, then $z$ and $w$ can be written as follows: the beginning z're'i'theta' quad and quad-core. endeqnarray Then the equation (reflog1) becomes $$e ueiv're'i're'theta'.$$$$$$$$$$$$$ So we have the start e'u'u'quad'text and quad vTheta2n'pi end eqnarray there, where $n Since $e'u'r$ is the same as $u'ln r$, it follows that the equation (reflog1) is satisfied if and only if the $w$ has one of the values of start w'ln r qy (Theta 2n'pi). Thus, the (multi-communic) logarite function of the non-zero-complex variable $z're'i'Theta'$ is defined by the formula starteqnarraylabel log log log r i'left) (Theta 2n'pi (right) Example 1: Calculate the $Log z$ for $z-1-sq{3}). Solution: if $z-1- sqrt{3}i$, the $r-2$and $-Theta'-frac-2'{3}${3}. -Frac'2'pi'{3}'2n'pi'right) {1}{3})pi i$$ from $nin mathbb .$ received from the equation (reflog2), when $n and $0, and is marked $text Log, z$. Thus, $$text (Log), z'ln r zita .$$ Feature $'text,Log, z$ well defined and one price when $zeq $0 and that $$'log z'text,Log', z'2n'pi i'quad (n'in 'mathbb) $$ it comes down to the usual logarithm in calculus when $z is a positive number. Illustration 2: Calculate $1 (left)right)$ and $lyologist (left (-1) right)$. Solution: From expression (ref'log2) $$'log (left (1)1'i'left (0'2n'pi'right) 1'i'left (Pee- 2n'pi'right) left (2n'1'right)'pi i'quad (n'in 'mathbb) $$$$Text-Log, Log, (1) $0 and $'text (-1) y pi i$. Expression (ref'log2)) is also equivalent to the following: start ecnarray I, textbf-arg (z) i, textbfarg (z) 2ni, quad (n'in mathbb : $magazine left (z_1 z_2 z_2) magazine z_1 $Yon on the left (frak z_1 z_2) Log z_1 - magazine z_2$ There can spend $ (text) , left (z_1,z_2) eq (text, Log), z_1 z_2$ Branch Of Logarith From the definition (reflog2) allow $'theta Theta 2n'pi$ ($nin mathbb$), so we can write start eqnarray label log30 log z ln ri'theta. Now, let $'alpha$ be any real number. If we limit the cost of $'theta$, so that $'alpha zlt; theta zlt; alpha and 2n'pi$, then the function start eknarray label log zln r i'theta quad (r'gt; 0, alpha zlt; theta zlt. end'eqnarray with starteqnarray components u (r, theta)ln r, quad v(r,theta)theta, endeqnarray is a single-component and continuous function in the stated domain. The multi-cost $f feature branch is any one-to-one $F feature that is analytical in some domain areas at each point $z$, of which the value of $F (z) $ is one of the values of $f$. The analytics requirement, of course, does not allow $F $10 from randomly selecting values $f$. Note α for each fixed setting, the single-room function (reflog3) is a branch of the multi-part function (reflog30). Starteqnarraylabellog4 (text Logue z'ln r iTheta quad quad (r'gt; 0, -'pi zlt; theta zlt; pi), endeqnarray is called $F the main branch. Branch cut is a part of a line or curve that is introduced to determine the branch of $F $F$$f a multi-show function $f. is called the branch point. The origin and beam of $'theta and alpha$ make up a branch carved for the branch (ref'log3) of logaritic function. The branch carved for the main branch (ref log4) consists of origin and beam $ (pi$). Origin is obviously the point of the branch for the branches of the multi-lumpy logarithic function. We can visualize the multidimensional nature of the $Log z$ using Riemann surfaces. The following interactive images show real and imaginary components of $z)$. Each branch of the imaginary part is identified with a different color. Unfortunately, applet is not supported for smaller screens. Turn the device onto the landscape. Or miser the window to be wider than the high one. Particular attention should be taken when using the branches of logarithic function, especially since the expected identifiers with the participation of logariths are not always transferred from calculus. Please note that for $zeq $0, we have a beginning eqnarray exp001 e'log z'z'quad text and four (e'z) z2n'pi i endeqnarray with $n in mathbb .$ Example 3: Calculate $e log z$, and $1 million for $z $4i. Solution: If $z $4i, then $e z'z'e'4i'$. So $$'log'left (e'4i'right) 4i (2n'pi i$$ with $n'in .$ On the other hand, we have a $$e magazine left (4i-right) 4i.$$NEXT: Riemann Surfaces - Intro, Source, questions: 978-0-6485736-9 2019 Logarithm complex number of one branch of complex logarithm. , a complex logarithm non-zero complex number z, denoted w q log z, is defined as any complex number w for which e w z. The real exponential function e y, satisfying e ln x x for positive real numbers x. If a non-zero complex number of z is given in a polar form, both z and re iθ (r and θ real numbers, with r qgt; 0), then w0 and ln (r) iθ is one logarithm z. Since z re i (θ and 2k) exactly for all integer k, adding multiple numbers 2 to the argument θ gives all the numbers that θ are logarithms z: All these complex logarithms z are on the vertical line in a complex plane with a part of the rn). Since any non-grain-complex number has infinite numbers of complex logariths, complex logariths cannot be defined as a one-dimensional function on complex numbers, but only as a multivalent function. Settings for formal processing of this, among other things, related Riemann surfaces, branches, or partial reverses complex exponential function. Sometimes ln notation is used instead of a log when dealing with a complex logarite. Problems with inverting the complex exponential function Plot a multi-valuable imaginary part of a complex logarith function that shows the branches. As the complex number of z goes around the beginning, the imaginary piece of logarithm goes up or down. This makes the origin the point of the function branch. For the function to be reversed, it must match different values with different values, meaning it must be injected. But the complex exponential function is not injectable because ew'2'i ew is for any w, since adding iθ to w has the effect of rotating ew counterclockwise θ radian. So the dots ... w No 4 π i, w q 2 π i, w, w. π π . ldots, ' equally, separated along the vertical line, all displayed on the same number exponential function. This means that the exponential function has no reverse function in the standard sense. There are two solutions to this problem. One is to limit the area of exponential function in a region that does not contain any two numbers different from a number of 2'i: this leads, naturally, to the definition of z log branches, which are certain functions that allocate one logarithm of each number in their areas. This is similar to the definition of arcsin x at No.1, 1 as the reverse limitation of sin θ to the interval of π/2, π/2: There are infinitely many real numbers θ with sin θ and x, but one arbitrarily chooses one in π/2, π/2. Another way to solve uncertainty is to view the logarite as a function, the area of which is not the area in plane, and the surface of Riemann, which covers a punctured complex plane in an endless way. Branches have the advantage of being valued at complex numbers. On the other hand, the function on the surface of Riemann Riemann in that it packs together all logarithm branches and does not require arbitrary choice as part of its definition. Determining the main value for each non-grain-complex number of z and x y yi, the main value of Log z is logarithm, whose imaginary part lies in the interval (π, π). The log 0 expression remains uncertain as there is no complex number of w satisfying ew No 0. The main meaning can also be described in several other ways. To give a formula for Log z, start by expressing z in polar form, z and reiθ. Given z, the polar shape is not entirely unique, due to the possibility of adding a multiple of 2 to θ, but it can be unique, requiring θ lie in the interval (π, π); this θ is called the basic meaning of the argument, and is sometimes written by Arg z or (especially in computer languages) atan2 (y,x), which agrees with arctan (y/x) when x zgt; 0, but gives the correct value to any (x, y) ≠ (0, 0). Then the main value of logarithma can be determined by the magazine ⁡ z ln ⁡ r , i θ ln ⁡ q i arg ⁡ z y ln ⁡ x 2 y 2 i atan2 ⁡ (y, x). Displaystyle operator name Log z'ln r'i'theta ln zi'operatorname Arg z'ln sqrt x{2}y'{2}'i'operatorname (y,x). Another way to describe Log z is to reverse the complex exponential function, as in the previous section. The horizontal S band, consisting of complex numbers w and x'yi, is such that π zlt; y ≤ π is an example of a region that does not contain two numbers different from the 2'i multiple, so limiting the exponential function of S has a downside.
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