Dynamic Visualization of Complex Integrals with Cabri II Plus

Transcription

1 Dynamic Visualiation of omplex Integals with abi II Plus Sae MIKI Kawai-juu, IES Japan Abstact: Dynamic visualiation helps us undestand the concepts of mathematics This pape shows that of complex integals with abi II Plus The geometical intepetation of complex numbes was intoduced by Gauss using the complex plane, but in geneal this geometical idea has not been sufficiently made much of in the scene of education This is only taught in the beginning, and is aely used in teaching complex integation Futhemoe its dynamic visualiation is hadly found out And this is one of the most difficult fields fo students, while it often appeas in studying science This pape highlights not only the visualiation of mathematical concepts but also the effectuality of moving figues Obseving thei movements maes it easie to find out mathematical popeties abi Geomety enables us to mae moving figues without the nowledge of pogamming omplex integals ae visualied with the idea of the quadatue mensuation by pats, using the geometical intepetation of the poduct of complex numbes; otation and dilation This dynamic visualiation helps students undestand complex integation deeply Intoduction This pape gives the dynamic visualiation of complex integals with abi II Plus, which is a dynamic geomety softwae ([] The integal of a eal vaiable function y=f(x on the xy-plane is easily visualied as an aea This idea is often used in the scene of the education, and helps students undestand the meaning of the integation As fo the complex integation, such visualiation is hadly shown, and it may dive students to thin that complex integation is difficult Thee ae only some wos ([], [3], and [4] fo the visualiation of complex integals I d lie to expand thei wos In this pape we can get esults of integals without calculation We only need to daw a cicle (o an ac with a cetain length of a sting Futhemoe, in this way, we can find the symmetic popety of complex integals Thus we easily undestand the special popety of /, and it leads us to open the doo to the concept of esidues, which explains us the necessity of Lauent expansion abi II Plus povides us oppotunities to ceate moving figues only with basic nowledge of geomety

2 Visualiing Fou Opeations of omplex Numbes We will get the values of integals by epetition of multiplication and addition, which ae geometically visualied on Gauss plane Addition and subtaction of complex numbes ae shown as those of vectos We can use tools of abi II Plus to daw the sum of vectos As fo multiplication and division, they ae shown as otation and dilation; i =,ag = θ, ie, e θ = (,, 3 =, 3 = 3 =, θ3 = θ+ θ The maco in abi II Plus can emembe a flow of constuctions and is useful fo multiplication and quotient of vectos which need epeated constuctions In this pape we egad the aguments of complex numbes to be the positive least 3 Visualiing omplex Integals 3 Quadatue Mensuation by Pats Def f ( d = lim f( ( = + The integal is defined as the quadatue mensuation by pats, which is the sum of poducts Fo simplification, we divide the contou equally by n; all the s have the same length As fo the eal integation, it can be explained as the aea; b n n = 0 ( f ( xdx = lim f( x x = S: aea a n n = 0 { f f f } = lim ( + ( + L + ( n 0 0 n n 3 omplex Integals of α along the icle centeed at the oigin and adius (α : intege The following examples visualie complex integals by multiplication and addition as we have seen in the pevious chapte Th ( α ( α = d 0 α = πi : the cicle centeed at the oigin and adius, : the loop integal Ex ( α d = 0 =

3 def d = lim ( L + n n ( = + n With n lage enough, we can egad uuuu uuuuuu OP and PP + to be pependicula and the total sum of s to be the whole cicumfeence of the cicle Now let s thin about the esult of the poduct See Fig (This figue is constucted with the tools of abi II Plus such as icle, Pependicula Line, Measuement Tansfe, etc We can mae a maco to constuct diectly fom and Let n be 0, fo example π π π =, =, ag = = θ, ag ( = θ π π =, ag( = θ + ( = 0,, L, n 0 Fig The Result of the Poduct Then we tae the sum fo = 0,, L,9 See Fig (Macos made in abi II Plus help us epeat the constuctions 4 3 =0,,,, Fig Taing the Sum of s fo = 0,, L,9

4 Imagine a sting winded fom the oigin aound a cicle This sting has the length π 9 =, and goes aound anticlocwise along the cicle twice until the end of the whole = 0 9 π cicle, because the sum of the exteio angles equals to 4π = { ( θ+ θ } = 0 = 0 0 The esult of this calculation appeas as the end point of the sting; 0 Now we get Theefoe, 9 = 0 ( = 0 n= 0 ( α d = 0 = Ex Now let s thin about the esults of the poduct d = πi = ( α See Fig3 =, π =, n π ag = = θ n, ag ( π = θ + π =, n π π ag( = θ + θ + = ( = 0,, L, n Note that evey is independent of, and ag( always equals to π Fig3 The Result of the Poduct

5 Let n be 0, and we shall obseve the sum of s ( = 0,, L, n See Fig4 9 9 π i =0,,,, 0 0 Fig4 Taing the Sum of s fo = 0,, L,9 This time the sting has the length π, and goes upight The esult of this calculation appeas as the end point of the sting; π i Now we get 9 = π i ( n= 0 = 0 Theefoe, d = πi ( α = Γ In addition, this example has also visualied the following integal; d log = + Ex3 Now let s thin about the esult of the poduct See Fig5 d 0 = = ( α =, π =, ag π θ = = n n, ag ( π = θ + π π =, ag( = θ ( = 0,, L, n n

6 Fig5 The Result of the Poduct Fig6 Taing the Sum of s fo = 0,, L,9 This time the sting has the length π, and goes aound clocwise along the cicle once until the end of the whole cicle The esult of this calculation appeas as the end point of the sting; 0 See Fig6 Now we get

7 Theefoe, 9 = 0 ( n = 0 = 0 d = 0 ( α = Summay of 3 Substituting α = β into Th, we obtain the following theoem Th β ( β ( β = d 0 0 = πi 0 : the cicle centeed at the oigin and adius, : the loop integal As well as Ex to 3, def β β 0 β β n d lim 0 n = + + L + n 0 n β β =, π n ag = βθ, ag =, ( β π = β π =, ag( = βθ + n β β π The sting has the length π β, and goes aound anticlocwise along the cicle β times ( if β < 0 then clocwise, and if β = 0 then goes upight until the end of the cicle The esult of this calculation appeas as the end point of the sting; 0( β 0 and i ( 0 π β = See Table

8 The Length of the Sting = π β Table Popeties of the Sting Appeaing in β d n n n β β β π β π β ; the length of = = = n = π = 0 = 0 = 0 n n How Many Times the Sting Tuns Aound = β times anticlocwise n n β + β π ; the sum of the exteio angles = + = { β ( θ+ θ } = n β = πβ = 0 + = 0 n The End Point of the Sting = 0( β 0 and πi ( β = 0 The esults led by this method ae justified by the following calculations; iθ iθ = e, θ :0 π, d = ie dθ π π β β iβθ iθ β iβθ d = e ie dθ = i e dθ 0 i θ= θ e θ= 0 π β β iβθ πβ i e e e iβ = = = β θ = 0 i θ = 0 i 0 ( 0 ( β 0 π [ θ] = πi ( β = 0 These esults ae symmetic with espect to β Now we get the symmetic popety of integals See Table

9 β = 0 d = π i length = π Table The Symmetic Popety of Integals otaion = 0times i β = β = d = 0 length = π otaion = time d = 0 length = π otaion = times β = β = d = 0 length = π otaion times = d = 3 0 length = π otaion = times β = 3 β = 3 d= 0 3 length = π otaion = 3times d = length = π otaion = 3times

10 33 Lauent seies and Residue The discussion in 3 shows the following theoem Th3 f ( d = πia a a whee f( = L+ + + a0 + a+ a + L (* : the cicle centeed at the oigin and adius (* is called Lauent seies, and this is the theoem of esidue, which shows the impotance of Lauent expansion 4 onclusion Dynamic visualiation is so effective Thans to this idea, we can undestand the complex integals and find the vaious popeties In the pocess of visualiation I found the symmety of integals Integals along the othe contous ae also visualied in this way, and othe theoems, fo example Rouche s theoem, could be visualied with this idea I hope students ae taught the complex integals with this idea and easily undestand the concepts with images Fo dynamic visualiation abi II Plus is vey useful Refeences [] abi Geomety II plus, [] KOBAYASHI Ichio, Gasping the concept of integals of complex-valued functions using abi, IME9 abi poject, 000, [3] KOBAYASHI Ichio, Undestanding omplex Analysis concepts with the help of abi, abiwold00 in Monteal, 00, [4] Tistan Needham, Visual omplex Analysis, laendon pess, Oxfod, 997