Principal branch of log

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←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. In fact, the exponential function of the S bijectively map to puncture the complex plane C ∖ × × → In the section of conformal mapping below describes in more detail the geometric properties of this map. When a z notation log appears without any specific logarithm being indicated, it is generally best to assume that the main value is intended. In particular, it gives value according to the real value of ln z, when z is a positive real number. Capitalization in the journal of notations is used by some authors in order to distinguish the main value from other logarites z. Not all identities satisfied with ln extends to complex numbers. It is true that eLog z z for all z ≠ 0 (this is what it means for the magazine z to be logarithm z), but the personality of Log ez z fails to z outside the band S. For this reason, it is not always possible to apply the magazine to both sides of the identity ez and ew to deduce z q w. Also, the identity magazine (z1z2) - Journal z1 and Log2 can break out of order: two sides may differ from the integrator, multiple 2'i: for example, the entrance ⁡ ⁡ ⁡ ( ...... π π ...... {2} ...... Frak pi i{2}, but the magazine ⁡ (⁡ π No. 1) - magazine ⁡ (i) (ln ⁡ (π 1 π ≠ π). Displaystyle operator name Log (-1) operator name (i) left (1) piiright) on the left (1) frak pi{2} right) {2} frac 3'pi i {2} Log z function is intermittent with every negative real amount, but is continuous everywhere in C ×. To explain the gap, think about what happens to Arg z, as z approaches the negative real number a. If z approaches from above, then Arg z is approaching π, which is also the value of Arg a. But if z is approaching from below, then Arg z is approaching π. Branches of complex logarithma Is there another way to select the logarithm of each non-grain complex number in order to make the function L (z), which is continuous at all C × display mathbb C times ? The answer is no. To understand why, imagine tracking such a logarite function along the circle of a unit, estimating L on eiθ θ increases from 0 to 2. To be simple, let's assume that the original L(1) is 0. Then for L(z) to be continuous, L (eiθ) has to agree with iθ as θ increases (difference is the continuous function of θ taking values in discrete set of 2 π i displaystyle 2pipi'mathbb). Specifically, L(e2'i) is the 2nd, but e2'i 1, so it contradicts L(1) 0. To obtain a continuous logarite defined by complex numbers, you need to limit the domain to a smaller subset of U complex plane. Since one of the goals is to be able to differentiate the function, it is reasonable to assume that the function is determined by the proximity of each point of its domain; in other words, u should be an open set. It is also reasonable to assume that U is related, since otherwise the functions on different U components may not be related to each other. All of this motivates the following definition: the z log branch is a continuous function of L/z, defined on a connected open subset of U complex plane in such a way that L(z) is a z logarite for every z in U., for example, the value defines a branch on an open set where it is continuous, i.e. a set of C and R ≤ 0 mathbb C - mathbb R (R) obtained by removing 0 and all negative real numbers from a complex plane. Another example: The Mercator series ln ⁡ ( 1 + u ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n u n = u − u 2 2 + u 3 3 − ⋯ {\displaystyle {\displaystyle N1'u'u-frac (-1)n'u'n'u-frac (u'{2}) {2} frac (u'{3}){3}-cdots) converges locally evenly for zlt;1, so the z q 1'u setting defines a z magazine branch on an open disk radius 1 centered in 1. (In fact, this is just a limitation of Log z, as you can show, differentiate the difference and compare values by 1.) Once the branch has been corrected, it can be labeled as log z if there is no confusion. Different branches can give different values for a logarithm of a particular complex number, however, so the branch must be fixed in advance (otherwise the main branch must be understood) in order for the z log to have an exact unambiguous value. The branch cuts the argument above involving block circle generalizes to show that no branch of the z log does not exist on an open U set containing a closed curve that winds around 0. To break this argument, U is usually chosen as a beam or curve supplement in a complex plane, moving from 0 (inclusive) to infinity in some direction. In this case, the curve is known as a branch incision. For example, the main branch has a branch carved along the negative real axis. If the L(z) function is extended to be determined at the point of the branch incision, it is sure to be intermittent there; at best it will be continuous on the one hand, like Log z on a negative real number. Derivative of complex logarith, each branch of the magazine z on the open set of U is the reverse limitation of exponential function, namely the limitation of the U image under L. Since the exponential function is holomorphic (i.e. complex different) with non-vanish derivative, a complex analogue of the reverse function theorem is used. This shows that L (z) is holomorphic on every z in U, and L' (z) 1/z. Another way to prove it is to check the Cauchy-Riemann equations in polar coordinates. Building branches with integration Ln ⁡ (x) display (ln(x) for x zgt; 0 displaystyle x'gt;0 can be built by the formula ln ⁡ (x ) ∫ 1 x d u u . If the {1} integration range started with a positive number other than 1, the formula should be ln ⁡ (x) ln ⁡ (a) ∫ a x d u'displaystyle (x) When developing an analogue of a complex logarism, there is another complication: the definition of a complex and complex path requires a choice of path. Fortunately, if the integrand is holomorphic, then the integral value remains the same, distorting the path (while keeping the endpoints fixed), and in a simply connected U area (an area with no holes) any way from z inside you can be continuously deformed inside U in any other. 0, the branch of the magazine z identified on you can be built by selecting a starting point in U, selecting logarithm b, and the definition of L (z ) q b - ∫ a z 'displaystyle L'z' b'int a'z'frac dw'w for each z in U. 4 Complex logarithm as conformal map Circles Re (Log z) Any holomorphic card f : U → C displaystyle fcolon U'to mathbb (C) (z) ≠ 0 displaystyle f'(z)eq 0 for all z ∈ U displaystyle z in U is a conformal map, which means that if two curves passing through point U form the angle of the α (in the sense that the tangent lines to the curves at an angle α), the images of the two curves form the same angle α on f(a). Because the branch of the magazine z is holomorphic, and since its derivative 1/z is never 0, it determines the conformic map. For example, the main branch of w q Log z, seen as a display from C - R ≤ 0 (display) (mathbb) - Mathbb (R) (R) to the horizontal band defined by Im z z lt; π has the following properties that are the direct consequences of the formula in terms of polar form: Circles in z-plane, centered on 0, are displayed on vertical segments in w-plane, connecting with the qi. Beams coming from 0 in the z-plane are displayed on horizontal lines in the w-plane. Each circle and beam in the z-plane, as above, meet at right angles. Their images under the log are a vertical segment and a horizontal line (respectively) in the w-plane, and they also meet at right angles. This is an illustration of the conformity of the magazine. The bound surface surface of the Riemann Imaging surface of the Riemann journal z. The surface appears to spiral around a vertical line corresponding to the origin of a complex plane. The actual surface extends arbitrarily both horizontally and vertically, but is cut off in this image. The design of the various branches of the z magazine cannot be glued to give one continuous log of functions: C × → C (display style) (colon))mathbb (C) to mathbb C because the two branches can give different values at the point where both are defined. Compare, for example, the main branch of the Log (z) on C - R ≤ 0 (display style) mathbb (C) - mathbb R (R) with the imaginary part of the θ in (π.π) and the L(z) branch on C - R ≥ 0 (display style) mathbb (C) - Mathbb R (R) (R) whose imaginary part of the θ lies in (0.2). They agree on the top half of the plane, but not on the lower half of the plane. Therefore, it makes sense to glue the domains of these branches only along the copies of the top half of the plane. The resulting glued domain is connected, but it has two copies of the lower half of the plane. These two copies can be visualized as two levels of garage, and can be obtained from the magazine The lower half of the plane to the L level of the lower half of the plane, went 360 counterclockwise about 0, first crossing the positive real axis (log level) into the overall copy of the top half of the plane, and then crossing the negative real axis (level L) to the L level of the lower half of the plane. You can continue gluing branches with an imaginary part of the θ in (π.3), in (2.4), and so on, and in the other direction, branches with an imaginary part of the θ in (2.0), in (3, π), and so on. The final result is a connected surface, which can be seen as a spiral garage with an infinite number of levels extending both up and down. This is the Riemann R surface associated with the magazine z. Point on R can be seen as a pair (z,θ), θ is a possible value of the z argument. So R can be built into the C × R ≈ R 3 (display) mathbb (mathbb) (mathbb) (mathbb) (r) (approximately) mathbb (R) (R) ({3}). The logarite function on the surface of Riemann Because branch domains have been glued only along open sets where their values are aligned, the branches are glued to give one clearly defined log of functions R: R → C displaystyle magazine R colon R'to mathbb C . . It displays each point (z,θ) on R to ln z and iθ. This process of expanding the original branch of the journal by gluing together compatible holomorphic functions is known as analytical continuation. There is a projection map from R to C × mathbb C display times that aligns the spiral, sending (z,θ) on z. For any z ∈ C × displaystyle zin mathbb C (time), if one takes all points (z.θ) R lying directly above z and evaluates logR on all these points, then one gets all logarithms z. Glued all branches of the magazine z Instead to glue together only the branches selected above, you can start with all the branches of the z log, and simultaneously glue each pair of branches L 1 : U 1 → C display L_{1} colon U_{1} to mathbb and L 2 : U 2 → C display L_{2}U_{2} to mathbb (C) - on the largest open subset U 1 ∩ U 2 displaystyle U_{1} cap U_{2} on which L1 and L2 agree. This gives the same Riemann Surface R and logR function as before. This approach, although a little more difficult to visualize, is more natural in that it does not require the selection of any specific branches. If the U.S. is an open subset of R projected on its U image in C × displaystyle mathbb C (C) , then the logR restriction on U corresponds to the branch of the magazine z, defined on the U. Each branch of the magazine z arises in this way. Riemann's surface as a universal cover Projection card R → C × R display to mathbb C time implements R as a cover of space C × display mathbb C times. In fact, this Galois cover with a group conversion deck isomorphic to 'display style', generated Khomeomorphism sending (z,θ) in (z,θ '2). As a complex variety, R is bicholomorphic with C display mathbb C via logR. (Reverse card sends z to (ez,I'm z).) This shows that R is just connected, so R is a universal cover of C × display mathbb C times. The Complex logarithm application is needed to determine an exposure in which the base is a complex number. Specifically, if a and b are complex numbers with ≠ 0, you can use the primary value to define ab and eb Log a. You can also replace the log with other logarites A to obtain other ab values, as displaying w q Log z converts circles focused on 0 into vertical straight-line segments, which is useful in engineering applications involving annu. (quote is necessary) Logarithms generalizations on other bases Just as for real numbers, can be defined for complex numbers b and x magazine b ⁡ (x) - magazine ⁡ x magazine ⁡ b , display style magazine b (x) frac magazine x'log b , the only caveat is that its value depends on the choice of the branch of the magazine, defined in the ≠ magazine For example, i ⁡ (e) - e log ⁡ i 1 ⁡ π i/2 and 2 i π - uses the main value. Displaystyle you (e) frak frac magazine and frak {1} pi/2-frak 2ipi. Logarithms of holomorphic functions If f is a holomorphic function on a connected open subset of U C 'displaystyle (mathbb) , then the branch of the magazine f on U is a continuous function of g on U, so, for example, z)f(z) for all z in U. Such function g is necessarily holomorphic with gz (z) - f'(z)/f (z) for all z in U. If U is just a connected open subset of C ' displaystyle (mathbb) and f is nowhere disappearing holomorphic function on U, The f log branch, defined on you, can be built by selecting a starting point in U, choosing logarithma b f(a) and defining g ( z) b q b q ∫ a z f' f (w) d s'display Look also Arg (mathematics) Discrete logarithm Exhibitor Reverse Trigonometry Features Logarithm Notes Section IV.9. Conway, 39. Another interpretation of this is that the reverse complex exponential function is a multivalent function, resulting in each non-grain complex number of z in the set of all logariths z. Strictly speaking, the point on each circle of the negative real axis must be discarded, or the main value should be used there. R and logR notations are not used everywhere. Erwin Kreisig (August 16, 2011). Advanced Engineering Mathematics (10th (posthumous) Berlin: Wylie. page 640. ISBN 9780470458365. Links Conway, John B. (1978). The functions of one complex variable (2nd p.m. Springer. Lang, Serge (1993). Gino (1964). The functions are a complex variable. Prentice Hall. Sarason, Donald (2007). Complex function theory (2nd place). American Mathematical Society. Whittaker, E. T.; Watson, G.N. (1927). Course of modern analysis (fourth place). Cambridge University Press. Extracted from the principal branch of logarithm. principal branch of logz. principal branch of log function. computing the principal branch of log-gamma. principal branch of the logarithm for complex exponentiation

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