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the loewner differential equation was introduced by karl lwner  @xcite ( who later changed his name to charles loewner ) to study properties of univalent functions on the unit disk . the differential equation is driven by a function that encodes in an ingeneous way continuous curves slitting the disk from the boundary . a few years ago , oded schramm realized that by taking brownian motion as the driving function , the loewner equation generates families of random curves with a conformally invariant measure , that he called `` stochastic loewner evolutions '' ( sle s )  @xcite . moreover , he showed that if the loop - erased random walk has a scaling limit , and if this scaling limit is conformally invariant , then it must be described by an sle , and he made similar conjectures about the scaling limits of uniform spanning trees and of critical percolation . the existence of conformally invariant scaling limits and the connections with sle were established for the first two models by lawler , schramm and werner  @xcite , and for critical site percolation on the triangular lattice by smirnov  @xcite . over the past few years schramm s idea has been investigated further , mainly in a combined effort of lawler , schramm and werner  @xcite , leading to new developments in the study of ( the scaling limit of ) discrete models .    in sle an ensemble of driving functions generates an ensemble of shapes . it is a difficult task to understand fully the relation between these two ensembles . for this reason we investigate here the simpler problem of the relation between a driving term in loewner s equation and the curve it generates , by explicit calculation . in particular we study exact , deterministic solutions of loewner s equation in the upper half  @xmath13 of the complex plane . one of the purposes of this study is to present some explicit examples that may help to elucidate the behavior of the curves generated by loewner s equation . for example , our solution for square - root forcing in section  [ squarerootforcing ] shows a transition from a simple curve that stays above the real line to a curve intersecting the real line , which is similar to the transition occurring at @xmath14 in sle .    to get us started , we need to understand how the loewner equation describes curves slitting the half - plane  @xmath13 . so suppose that @xmath15 , @xmath16 , is a simple curve in @xmath13 emanating from the real line . then according to conformal mapping theory , there exists a unique conformal mapping of the form @xmath17 that takes the domain  @xmath18 consisting of all points in  @xmath13 minus those on the slit  @xmath19 $ ] onto the upper half of the @xmath20-plane , in such a way that near infinity this map has the expansion @xmath21 see figure  [ slitmap ] for a schematic picture of the mapping . it is a theorem that the coefficient @xmath22 ( which is called the _ capacity _ ) is continuously increasing with  @xmath6 . therefore , the parameterization of the curve can be chosen such that @xmath23 , and henceforth we assume that this is the case .     in the @xmath24-plane maps into a linear boundary in the @xmath20-plane . notice that two neighbouring points in the @xmath24-plane have images which are far apart in the @xmath20-plane . ] the conformal maps @xmath25 satisfy a surprisingly simple differential equation , which is the loewner equation    [ le ] @xmath26 with the initial condition @xmath27    for all @xmath24 in @xmath13 . here , the value of @xmath28 at time @xmath6 is just the image of the point @xmath15 under the map  @xmath25 . we call @xmath0 the driving function or forcing function of the loewner equation . it is continuous and real - valued . ( for a text - book discussion of loewner s equation and proofs of the above statements when the domain is the unit disk , see @xcite and  @xcite . the half - plane case we are considering here is analogous and is discussed in  @xcite and  @xcite ) . conversely , the loewner equation  ( [ le ] ) has a solution for any given continuous real - valued function  @xmath0 . this generates the function  @xmath29 . for each value of @xmath6 this function can be thought of as a mapping of the form @xmath17 which takes some connected subset of the upper - half @xmath24-plane , @xmath18 , into the entire region  @xmath13 above the real axis of the @xmath20-plane . correspondingly , there exists an inverse function  @xmath30 , which obeys @xmath31 for all  @xmath20 in the upper half - plane . this function maps the region  @xmath13 of the @xmath20-plane into the region  @xmath18 of the @xmath24-plane . it obeys the partial differential equation @xmath32    the loewner equation continually generates new singular points of  @xmath25 that are mapped by  @xmath25 onto the corresponding points  @xmath0 in the @xmath20-plane . thus , as time goes on , points are continually removed from  @xmath18 . if  @xmath0 is sufficiently smooth  @xcite , these singular points form a simple curve  @xmath33 , where the points  @xmath33 obey @xmath34 this curve is called the trace of the loewner evolution , but we also refer to it as the line of singularities . ( naturally , if we take  @xmath0 to be the driving function corresponding to a given curve  @xmath15 as described above , then @xmath33 will coincide with  @xmath15 ) . equations  ( [ le ] ) have a simple scale invariance property under changing both the scale of space and of time . any change of time of the form @xmath35 can be compensated by a change of scale . to do this , define a new  @xmath36 and  @xmath28 by @xmath37 this pair then satisfies equations  ( [ le ] ) . note that by the schwarz reflection principle of complex analysis  @xcite , the map  @xmath25 extends to the lower half - plane and satisfies @xmath38 . it follows from equation  ( [ scale ] ) that flipping the sign of the forcing has the effect of reflecting the regions @xmath18 and  @xmath13 ( and hence also the line of singularities  @xmath33 ) in the imaginary axis . likewise , a shift in the driving function  @xmath0 can be compensated by a similar shift in  @xmath25 , since the pair @xmath39 again solves equations  ( [ le ] ) . this shows that a shift in the driving function simply produces a shift of the same magnitude in the trace  @xmath33 . in this note we are interested in the question how the behavior of the trace is related to that of the driving function . this question can be investigated both numerically and , in some cases , by exact methods . a simple numerical method to find the trace generated by a given driving function  @xmath0 , is to numerically integrate loewner s equation backwards from the initial condition @xmath40 to obtain the value of @xmath41 . to deal with the singularity in loewner s equation , we can use the first - order approximation @xmath42 substituting @xmath40 and solving this equation for @xmath43 gives an approximation for the value of  @xmath36 at time  @xmath44 , from where we then integrate backwards to obtain  @xmath45 . this method is good enough for the purpose of producing pictures . the figures [ linear ] , [ spirals ] , [ intersections ] and  [ doubletrace ] in this note were obtained in this way . exact solutions of equations  ( [ le ] ) can be found in a few cases . the simplest are ones in which the forcing has one of the forms    @xmath46    or @xmath47    for @xmath48 , @xmath2 , and @xmath3 . for a driving function of the form of equation  ( [ forceb ] ) , a rescaling according to  ( [ scale ] ) gives the new forcing @xmath49 this result allows us to scale away the constant  @xmath50 when @xmath51 . for the special value @xmath52 the multiplicative constant can not be scaled away , and can be important in determining the form of the solution .    in sections [ constantforcing ] , [ linearforcing ] and  [ squarerootforcing ] we shall derive the solutions respectively for the powers @xmath1 , @xmath3 , and  @xmath2 . section  [ halfcircle ] contains an explicit construction of the maps  @xmath25 for a half - plane slit by an arc , and we point out that this is a special case of square - root forcing with a finite - time singularity . in section  [ varyingforcing ] we consider a more general version of loewner s equation , where the forcing is a superposition of the values  @xmath12 and produces two lines of singularities . here we look at the almost trivial situation in which @xmath48 and we can take @xmath53 this situation is well - known ( see for example  @xcite and appendix  a in  @xcite ) , but for the sake of completeness we also treat it here . we could of course shift  @xmath54 away by using equation  ( [ shift ] ) , but this case is so easy that shifting is hardly worth the effort . the equation for  @xmath36 can be solved by inspection , giving @xmath55^{1/2}.   \label{gc}\ ] ] the inverse map is of the same form as the direct one , namely @xmath56^{1/2}.\ ] ]    at time  @xmath6 , the function  @xmath36 acquires a new singularity at the point @xmath57 which maps under  @xmath25 to the point  @xmath58 in the @xmath20-plane . thus the singularities form a line segment parallel to the imaginary axis and the mapped region of the @xmath24-plane is the upper half - plane minus that line segment . the mapping has a singular point at the place where the trace meets the real axis . we can approach this point @xmath59 from two different directions , giving us two different images at the points @xmath60 figure  [ constant ] gives an illustration of the mapping .        because any smooth enough function  @xmath0 looks locally as if it were constant , we might expect that smooth  @xmath0 s generate traces of singularities which look like simple curves coming up from the real axis . in fact , one can prove that this statement is correct whenever  @xmath0 is sufficiently smooth so that it is hlder continuous with exponent  @xmath2 and sufficiently small norm , i.e.  so that @xmath61 for all  @xmath6 and some constant  @xmath50 . the tricky issue is , when can it be that the curve will touch itself or hit the real line . condition  ( [ cond ] ) is enough to ensure that this will not happen , as was shown by marshall and rohde  @xcite , who did not give the value of the constant  @xmath50 . our solution in section  [ squarerootforcing ] below gives a specific example of the transition occuring at @xmath62 . in a parallel and independent paper , lind @xcite proves also generally that hlder-@xmath2 functions with coefficients @xmath63 generate slits . the next case , not much harder , has the forcing increase linearly with time according to @xmath64 we do not need an additive or multiplicative constant in equation  ( [ linear ] ) because these can be eliminated by equations ( [ scale ] ) and  ( [ shift ] ) . if we now redefine the independent variable to be @xmath65 , then  @xmath66 obeys @xmath67 where  @xmath68 is the function @xmath69 in terms of @xmath70 the solution is @xmath71 here , @xmath72 is a constant which must be set from the initial condition , @xmath73 , giving the solution @xmath74 or equivalently @xmath75    to find the trajectory of the singularity , use equations ( [ sing ] ) and ( [ csoln ] ) to get @xmath76 it is possible to find an explicit expression for this trajectory . to do so , substitute @xmath77 into the previous equation . then split the equation in its real and imaginary parts to obtain    @xmath78    after substituting the second equation into the first , it can be shown that  @xmath79 increases monotonously in time from the value @xmath80 to @xmath81 .    in terms of the parameter @xmath79 , the line of singularities may be written explicitly as @xmath82 this shows that the line of singularities moves outward to infinity while remaining within a fixed distance of the real axis . figure  [ linear ] shows the first part of this trajectory . -plane that arises from the forcing @xmath11 , up to time @xmath83 . ]    for small and large values of  @xmath6 we can also construct asymptotic forms of the solution  ( [ csoln2 ] ) . as  @xmath6 goes to zero , one can expand @xmath84 to third order in @xmath33 and find that , to this order , the result is @xmath85 for large  @xmath6 , @xmath86 + 2\pi i + o(\ln t / t).\ ] ] the next case , much more interesting , has the forcing be a square - root function of time . this case has two subcases : one in which the forcing has a finite - time singularity based on the rule of equation  ( [ forcea ] ) , i.e. @xmath87^{1/2 } , \qquad t\leq1 , \qquad \kappa\geq 0 . \label{forceap}\ ] ] here , as we shall see , the result changes qualitatively as  @xmath7 varies , with the critical case being @xmath14 where the singular curve intersects the real axis . in the other situation , the forcing has an infinite - time singularity with the rule @xmath88^{1/2 } , \qquad \kappa\geq 0 . \label{forcebp}\ ] ] in this case @xmath33 is just a straight line , as is already clear from the scaling relation  ( [ scale ] ) .      to find the solution for the forcing  ( [ forcebp ] ) , define the new variable @xmath89 and set @xmath90 . then  @xmath91 satisfies @xmath92 where @xmath93 . it follows that @xmath94    = \frac{1}{2}(y_+ - y_-).\ ] ] therefore , if we set @xmath95 then @xmath96 which integrates to @xmath97    the constant  @xmath72 can be determined from the observation that in the limit when  @xmath6 approaches  @xmath1 , @xmath98 our solution  ( [ forcebpsoln ] ) for general  @xmath6 becomes simply @xmath99 since the line of singularities is determined by the condition that  @xmath36 is equal to the forcing we get @xmath100.\ ] ] more explicitly the coefficient is @xmath101,\ ] ] so that the line of singularities is set at an angle to the real axis which is @xmath102 for @xmath103 the line is ( as we know ) perpendicular to the real axis while as @xmath104 the angle of intersection becomes smaller and smaller . now we turn to the forcing  ( [ forceap ] ) with a singularity after a finite time . in this case , one can eliminate the time - dependence by using the new variable @xmath105 then  @xmath91 obeys @xmath106 where @xmath107 . once again the derivative becomes a simple function of the unknown , again appearing in the form of a ratio of polynomials , i.e. @xmath108 where this time the roots are @xmath109 . notice that now the roots are real and positive when @xmath110 but that they are complex for @xmath8 . thus we can expect a qualitative change in the solution at @xmath14 . the integration of the equation proceeds exactly as in the previous case , giving the solution @xmath111 where  @xmath112 has the same form as before , but with different  @xmath113 : @xmath114 the initial condition then implies that @xmath115 and the equation for the line of singularities is then @xmath116 note that equation  ( [ sqrtsoln ] ) for the line of singularities tells us that @xmath117 must approach  @xmath118 when  @xmath6 approaches  @xmath3 . in the following subsections we shall consider the asymptotics of  @xmath33 in this limit in more detail . we separate the discussion into the two cases @xmath8 and @xmath9 . then the section is completed with the derivation of the shape of the critical line of singularities at @xmath14 . when @xmath8 , the real and imaginary parts of  @xmath119 are    @xmath120    from which it follows easily that @xmath121 is a positive real constant . since the real part of @xmath117 must go to  @xmath118 when  @xmath6 approaches  @xmath3 , we conclude that @xmath122 , since the other possible candidate for the limit , i.e.  @xmath123 , is in the wrong half - plane . it then follows from the observation that @xmath124 must vanish , that the trace must have wrapped around  @xmath125 an infinite number of times . hence the line of singularities is spiralling in towards the point  @xmath125 , as is depicted in figure  [ spirals ] .    . also shown is the limiting case @xmath14 where the trace starts to hit the real line . all the curves have been translated to make them start in the origin at time  @xmath126 . ]    to find a more explicit expression for the asymptotics , notice that when @xmath33 is close to  @xmath125 , we can make the approximations @xmath127 splitting equation  ( [ sqrtsoln ] ) in its real and imaginary parts , and substituting the above approximate values then gives a system of two equations that we can solve for the unknows @xmath128 and @xmath129 . this gives the result    @xmath130 / 4 , \\   { \mathop{\rm arg\,}\nolimits}(z_c(t)-y_+ )    & \approx & \big [ b(\kappa ) - \kappa^{1/2}(4-\kappa)^{1/2}\ln(1-t ) \big ] / 4,\end{aligned}\ ] ]    where the constants are @xmath131    thus , the distance between @xmath33 and  @xmath125 is decreasing like a power law in  @xmath6 , whereas the winding number of @xmath33 around  @xmath125 grows only logarithmically . the line of singularities intersects the real line at an angle that varies with @xmath7 . the curves were translated to make them start in the origin . ]    for @xmath9 the real and imaginary parts of  @xmath119 are    @xmath132    and @xmath121 is again a positive real number . this time , as  @xmath6 approaches one , @xmath33 must approach the point  @xmath123 on the real line , see figure  [ intersections ] . in fact , we can calculate the angle at which the line of singularities intersects the real line from the observation that @xmath124 must vanish .    indeed , if we denote by  @xmath79 the angle at which  @xmath33 hits the real line ( measured from the real line in the positive direction ) , then we have that in the limit @xmath133 , @xmath134 whereas @xmath135 . the condition @xmath136 then gives @xmath137 when  @xmath6 goes to  @xmath3 . as we can see , we have a glancing incidence for @xmath14 and @xmath104 , while for @xmath10 the incidence is perpendicular . in fact we can prove that for @xmath10 the line of singularities is a half - circle of radius  @xmath138 . we will do this in section  [ halfcircle ] , where we give an explicit construction of the conformal maps  @xmath25 when the half - plane is slit by a half - circle . it is not difficult to find an explicit expression for the critical line of singularities at @xmath14 . in this case , the two roots in equation  ( [ geqn ] ) are the same , @xmath139 , and the equation can be rewritten as @xmath140\frac{dg}{d\tau}=1.\ ] ] integration is straightforward and gives @xmath141 where we have determined the integration constant from the initial condition @xmath142 . the line of singularities is determined by the condition that for @xmath143 , @xmath144 .    substituting @xmath145 and splitting the equation in real and imaginary parts leads to the expression @xmath146 for the critical curve in terms of the parameter  @xmath79 . this parameter  @xmath79 increases monotonously with time from @xmath80 to @xmath147 . taking  @xmath13 minus the slit @xmath148 $ ] onto  @xmath13 , in three steps . the illustration also tracks the images of the upper half of the unit disk and of the four `` special '' points @xmath149 , @xmath1 , @xmath150 and  @xmath151 . the definitions of @xmath150 , @xmath152 and  @xmath153 are in the text . ]    in this section , we consider the situation where the upper half - plane  @xmath13 is slit by the curve @xmath154 , @xmath155 . the idea is to construct the conformal maps that map the half - plane minus the slit @xmath148 $ ] back to the half - plane and have the expansion of equation  ( [ expansion ] ) . after reparameterizing time one can then identify the driving function , and verify that the obtained maps  @xmath25 satisfy loewner s equation . the form of the normalized maps and of the driving function already appears in appendix  a of  @xcite , but for completeness we present a short account of the derivation of these results below . a simple rescaling argument then allows us to make the connection with section  [ squarerootforcing ] and prove that the square - root forcing we considered there generates a half - circle of radius  @xmath138 when @xmath10 .    to obtain the normalized conformal maps , we start by constructing the map  @xmath156 of figure  [ halfcircle ] as the composition of the three maps depicted there . the explicit form of this map is @xmath157\ ] ] where @xmath158 and the function  @xmath159 is given by @xmath160 with @xmath161 . we can now expand  @xmath156 for large  @xmath24 , and identify the capacity  @xmath162 . a simple translation will then give us the map satisfying  ( [ expansion ] ) . expanding  @xmath163 for large  @xmath24 up to order  @xmath164 gives @xmath165 therefore , the map @xmath166 has the expansion of equation  ( [ expansion ] ) , with capacity @xmath167 . the proper time reparameterization that makes the capacity equal to  @xmath168 is given by @xmath169 which has the inverse @xmath170 the maps  @xmath25 are then given by @xmath171 . more explicitly , this can be written in the form @xmath172 we know from figure  [ halfcircle ] that this map takes the point @xmath173 onto the image @xmath174 , which implies that the driving function must be @xmath175 indeed , one can check by differentiating  ( [ circlemap ] ) with respect to time , that  @xmath25 satisfies loewner s equation with this driving function .    to make the connection with section  [ squarerootforcing ] , we note that a rescaling of the form  ( [ scale ] ) with @xmath176 turns the driving function into a function of the form of equation  ( [ forceap ] ) with @xmath10 , up to a translation . we conclude that this particular square - root forcing produces a line of singularities that is a half - circle of radius  @xmath138 . we conclude this section with a discussion of its relation to an old paper by kufarev  @xcite . kufarev considers the equivalent of loewner s equation  ( [ le ] ) in the unit disk , and gives an explicit solution of this equation for negative times . a half - plane version of kufarev s example is obtained as follows . suppose that we set @xmath177 for  @xmath178 . then it can be verified that the solution of  ( [ le ] ) at time  @xmath179 is the normalized map that takes  @xmath13 onto @xmath13 minus the half - disk of radius  @xmath180 centered at  @xmath181 . observe that for  @xmath182 , up to a translation , this solution is just the inverse of  ( [ circlemap ] ) at time  @xmath183 . this is not a coincidence because generally , if one sets @xmath184 and solves loewner s equation for both @xmath185 and  @xmath186 , the two results are related by @xmath187 . in this section we look at a special case of the more general version of loewner s equation ( appearing for example in  @xcite ) @xmath188 where  @xmath189 is a measure on the real line that can be time - dependent . here we will take  @xmath189 to be time - independent and assigning a mass  @xmath190 to the values  @xmath191 such that @xmath192 . then the equation for  @xmath36 takes the form @xmath193 this can be seen as a case where the forcing  @xmath0 is a superposition of values , and we also believe that this can be described as a case where the forcing  @xmath0 makes rapid ( random ) jumps between the values  @xmath191 .    equations like  ( [ jump ] ) are generally easily integrated . take the specific case in which the possible values of  @xmath28 are  @xmath12 , which are taken on with equal probability . then equation  ( [ jump ] ) becomes @xmath194 which then integrates to give @xmath195 there are two traces , starting respectively from @xmath196 and then moving upward towards  @xmath197 . the traces are found , as before , by setting  @xmath36 equal to the forcing . thus the right - hand trace obeys @xmath198      again we can obtain an explicit expression for the trace , like we did for the case of linear forcing in section  [ linearforcing ] , and for the critical curve in section  [ squarerootforcing ] . after substituting @xmath200 and splitting the previous equation in real and imaginary parts , we obtain @xmath201 where  @xmath79 increases with time from the value @xmath80 to @xmath202 . note that we have filled in the results for both traces , using the fact that they must be symmetrically placed about the imaginary axis . we can also compute the asymptotics . for small values of  @xmath6 we obtain @xmath203 while for large values of  @xmath6 the traces behave like @xmath204 we see that as the traces move upward , they also approach one another , as is shown in figure  [ doubletrace ] . one of us ( lpk ) would like to thank leiden university and professor wim van saarloos for their hospitality for a part of the period in which the work was performed . we have had useful conversations about this research with paul wiegmann , ilya gruzberg , isabelle claus , marko kleine berkenbusch , and john cardy . we thank steffen rohde and joan lind for sharing their recent work on loewner s equation with us . lpk would like to acknowledge partial support from the us nsf s division of materials research . wk is supported by the stichting fom ( fundamenteel onderzoek der materie ) in the netherlands . 10 l.  v.  ahlfors , _ complex analysis : an introduction to the theory of analytic functions of one complex variable_. mcgraw - hill , new york , second edition ( 1966 ) . l.  v.  ahlfors , _ conformal invariants : topics in geometric function theory_. mcgraw - hill , new york ( 1973 ) . m.  bauer , d.  bernard , _ conformal field theories of stochastic loewner evolutions_. commun . 239 , pp . 493 - 521 ( 2003 ) ; arxiv : hep - th/0210015 . p.  l.  duren , _ univalent functions_. springer - verlag , new york , 1983 . an overview of the literature on schramm s loewner evolutions ( sle ) is given in ilya a.  gruzberg and leo p.  kadanoff , _ the loewner equation : maps and shapes _ ; arxiv : cond - mat/0309292 . kufarev , _ a remark on integrals of loewner s equation _ , doklady akad . nauk sssr 57 , pp . 655656 ( 1947 ) , in russian . g.  f.  lawler , _ an introduction to the stochastic loewner evolution _ , available online at url + http://www.math.duke.edu/@xmath205jose/papers.html g.  f.  lawler , o.  schramm , w.  werner , _ values of brownian intersection exponents i : half - plane exponents_. acta math . 187 , 2 , pp .  237273 ( 2001 ) ; arxiv : math.pr/9911084 . g.  f.  lawler , o.  schramm , w.  werner , _ conformal invariance of planar loop - erased random walks and uniform spanning trees_. ann . ( to appear ) ; arxiv : math.pr/0112234 . j.  r.  lind , _ a sharp condition for the loewner equation to generate slits _ ; arxiv : math.cv/0311234 . k.  lwner , _ untersuchungen ber schlichte konforme abbildungen des einheitskreises , i_. math .  ann . 89 , pp .  103121 d.  e.  marshall and s.  rohde , _ the loewner differential equation and slit mappings _ , available online at url + http://www.math.washington.edu/@xmath205rohde/ o.  schramm , _ scaling limits of loop - erased random walks and uniform spanning trees_. israel j.  math . 118 , pp . 221288 ( 2000 ) ; arxiv : math.pr/9904022 . s.  smirnov , _ critical percolation in the plane : conformal invariance , cardy s formula , scaling limits_. c.  r.  acad . .  i.  math . 333 , 3 , pp . 239244 ( 2001 ) . a longer version is available online at url http://www.math.kth.se/@xmath205stas/papers/
in this note , we solve the loewner equation in the upper half - plane with forcing function  @xmath0 , for the cases in which @xmath0 has a power - law dependence on time with powers @xmath1 , @xmath2 and  @xmath3 . in the first case 
 the trace of singularities is a line perpendicular to the real axis . in the second case 
 the trace of singularities can do three things . if @xmath4 , the trace is a straight line set at an angle to the real axis . if @xmath5 , as pointed out by marshall and rohde  @xcite , the behavior of the trace as  @xmath6 approaches  @xmath3 depends on the coefficient  @xmath7 . 
 our calculations give an explicit solution in which for @xmath8 the trace spirals into a point in the upper half - plane , while for @xmath9 it intersects the real axis . 
 we also show that for @xmath10 the trace becomes a half - circle . the third case with forcing  @xmath11 
 gives a trace that moves outward to infinity , but stays within fixed distance from the real axis . 
 we also solve explicitly a more general version of the evolution equation , in which  @xmath0 is a superposition of the values  @xmath12 . 
 [ [ key - words ] ] key words : + + + + + + + + + +    conformal map , loewner equation , singularity , slit domain .