Properties of generalized trigonometric functions

Transcription

1 University of West Bohemia Faculty of Alied Sciences Deartment of Mathematics Proerties of generalized trigonometric functions Bachelor thesis Pilsen 6 Martina Vašinová

2 Declaration I hereby declare that the entire bachelor thesis is my original work and that I have used only the cited sources. In Pilsen......

3 Acknowledgement I would like to thank Doc. Ing. Petr Girg, Ph.D., the suervisor of the thesis, for his valuable advice and exert guidance. I would also like to thank Ing. Jan Čeička, Ph.D., for useful consultation of the imlementation art of the thesis. And finally I am very grateful to Radovan Šrámek for a lot of grammatical corrections.

4 Abstract The main goal is to summarize known roerties of generalized trigonometric functions and aly them on comutation of integrals with these functions. Other imortant arts are code imlementation in Matlab, visualization of the functions and analysis of the numerical art of the roblem.

5 Contents Introduction. Analytic oint of view. Boundary and corresonding initial value roblem 3.3 Generalized trigonometric functions with two arameters 5.4 Structure of the thesis 5 Grahs 6 3 Integrals with generalized trigonometric functions 3. Basic substitutions 3. Further comutations 3.3 Functions with two arameters 3 4 Numerical art 4 4. Runge-Kutta method 4 4. Matlab ode solvers comarison 5 5 Imlementation 7 5. Values and grah of sin x and cos x 7 5. Values and grah of π,q 6 Conclusion 4

6 Introduction. Analytic oint of view A trigonometric function sin x is a π-eriodic function bounded by and. The inverse function, arcsin x, is described by the following formula: arcsin x = x We can generalize this function for < < : arcsin x = x which is an injective function from, to, π. We will describe π by the following equalities: dt, x. () t dt, x, () t dt = [arcsin t t] = π π = dt, t π = dt. (3) t We can tell from the relation above that π = π. The inverse function to arcsin x on, π is sin x. Let sin x be extended on x π, π symetrically: sin x = sin (π x). The function sin x is odd, that extends it for x π, and a π -eriodic function, that defines it for whole R. As we can see, sin x = sin x. These definitons for sin x and π are analogous to Edmunds and Lang s [5] and to Elbert s [6]. But the definition of cos x is different: Edmunds and Lang s [5] definition is and Elbert s [6] definition is cos x def = d dx sin x (4) cos x def = π x sin y dy. (5) In this thesis the first definition will be used. The function cos x is even and π -eriodic and cos x = cos x. We will define tan x just as tan x is defined: tan x def = sin x cos x, (6)

7 then tan x is odd and π -eriodic and tan x = tan x. We have to mention that tan x is not defined for x = (k + )π, k Z. The definitions of functions sec, cosec and cotan follow: cosec x sec x cotan x def = def = def = sin x, (7) cos x, (8) tan x. (9) Now we can examine if some well known roerties of sin x, cos x and tan x fit for generalized trigonometric functions too: sin ( x) = sin x, x R (sin x is odd) () cos ( x) = cos x, x R (cos x is even) () tan ( x) = tan x, x R (tan x is odd) () sin x + cos x =, x R (sin x and cos x are symmetric and eriodic) (3) sec x tan x =, x R \ (k + )π, k Z (4) We will rove (4): Let x R \ (k + )π, k Z: sin x + cos x = cos x = sin x. By dividing the equation by cos x and suosing cos x we obtain = sin x cos x = cos x tan x sec x tan x =.. Boundary and corresonding initial value roblem Let us consider following boundary value roblem: ( u u ) + λ u u =, (5) u() = u(π ) =, (6) where >, λ > are given real numbers. We will show the origin of arcsin (x) and sin (x) by rewriting (5): ( u ) u + λ u uu = ( ) u u u + λ u uu = ( u (x) u () ) + λ ( u(x) u() ) = u (x) + u(x) λ = 3

8 We assume λ =. u (x) + u(x) = We assume x [ ], π. (u (x)) + (u(x)) = t Let us substitute y = u(x), dy = u (x)dx. u(t) u (x) = ( u (x)) u (x) ( u (x)) = u (x) ( u (x)) dx = t ( y ) dy = t, which we have already mentioned in (). Let us find the corresonding initial value roblem to (5)-(6). We will aly following substitution: v = u, Obviously sgn(w) = sgn(v ), then: w = u u = v v = v sgn(v ). w = v w = v v = w sgn(w). By substitution w = u u in (5) we obtain: From boundary conditions (6) we obtain: w + λ v v = w = λ v sgn(v). v() =, w() = α, where α must be chosen according to condititon v(π ) =. This requirement meets α =. Previously we have asuumed λ =. The final corresonding initial value roblem follows: v = w sgn(w), (7) w = ( ) v sgn(v), v() =, w() =. Using Matlab function ode45 we solved Equation (7). 4

9 .3 Generalized trigonometric functions with two arameters In this thesis we will also consider sine and cosine functions with two arameters, and q. A version of the boundary value roblem (5)-(6) for two arameters follows: ( u u ) + λ u q u =, (8) u() = u(π,q ) =, (9) where >, q >, λ > are given real numbers. Using stes according to Section. we obtain following corresonding initial value roblem: v = w sgn(w), () w = λ v q sgn(v), v() =, w() =. Takeuchi [] showed, that sin,q x is the solution of (8) for λ = q( ). Definition of π,q follows:.4 Structure of the thesis def π,q = dt. () t q In Chater, we resent grahs of sin x, whose origin is based on (5)-(6), and cos x. We will also introduce π,q, whose value we found by exeriment using (). In Chater 3, we study integrals of generalized trigonometric functions and assemble them into the form of classical "Table of Integrals" by e.g. Gradshteyn and Ryzhik [7] and/or Prudnikov, Brychkov, Marichev [], []. In Chater 4, we make a remark about the numerical art of the roblem and in Chater 5, we introduce our Matlab codes, which were a significant art of the research. 5

10 Grahs We will introduce figures of our Matlab codes. As we know from Matlab documentation [9], the lot function connects oints by line segments. This fact could affect final grah. We will focus on Matlab codes and comutation of values of π,q, sin x and cos x in the Imlementation section. As we can see, the curve of sin x has secific shae for aroaching one and infinity [3]. Figure : sin x, on the left, = in between, on the right Figure : sin x 6

11 Figure 3: cos x It is not a surrise that for different there is a different eriod of the trigonometric function. Generally, sin x and cos x have eriod π. For the visualization of sine and cosine function above we have π 6 = π, π 3. = π and of course π 3 = π. In the code below you can see the comutation in Matlab as well as the code for lotting π. We used the formula from (3). Listing : Values and grah of π syms x; format long i6=*int(sf(6),x,,) %i_ definition, =6 i=*int(sf(.),x,,) %i_ definition, =. i=*int(sf(),x,,) %i_ definition, = Y=zeros(,37); %matrix will be used for saving i_ values i=; f o r =:.5: Y(,i)=*int(sf(),x,,); i=i+; end axis=:.5:; %values on -axis axis([ 3.]); hold on; lot(axis,y(,:),'black--o','linewidth',.),xlabel(''),ylabel('\i_'); 7

12 Listing : Function sf f u n c t i o n fun = sf() %suort function for comutation of integral in i_ definition syms x; fun=/(( - x^)^(/)); end We used codes in Listing and Listing to comute several values of π and to visualize π as a function of. You can see the final grah in Figure 4 below. Figure 4: π As we can see from the grah above, lim π = and lim π =, see [5]. In this context, there is an interesting fact about generalized sine and cosine function. We know that sin( π) = cos( π ). But for generalized sine and cosine the equation does not hold true. We 4 4 can see it in Figure 5, where we chose = 6, and in Figure 6, where we chose =.. The values of sine and cosine in π are circled in the grah. We used format long with 5 digits after 4 the decimal oint and we obtained following values: sin 6 ( π 6 4 ) = , cos 6 ( π 6 4 ) = , sin. ( π. 4 ) = , cos. ( π. 4 ) =

13 Figure 5: sin 6 x and cos 6 x Figure 6: sin. x and cos. x In figures above we set the scale of the x-axis to π. According to this we can get an idea of the intersection oint of sine and cosine functions. The exact comutation could be an object of further research. 9

14 Figure 7: π,q with fixed values of q In Figure 7 we can see a grah of π,q as a function of q. This means, that we fixed the value of and lotted the values of π,q for q {,.5, 3, 3.5, 4, 4.5, 5, 5.5, 6}. Figure 8: π,q with fixed values of In figure 8 there is a grah of π,q as a function of. We found the values of both grahs above by exeriment, which will be fully described in Chater 5.

15 3 Integrals with generalized trigonometric functions While solving integrals with trigonometric functions we usually use substitution. There are simle rules how to substitute in the table below. Table : Basic substitutions Tye of integral Substitution z dz dx dz f(sin x) cos x dx sin x cos x dx cos x f(cos x) sin x dx cos x sin x dx dz sin x f(tan x) cos x dx tan x dx cos x dz cos x We will study whether these methods are alicable for integrals with generalized trigonometric functions. Firstly, we will find solutions of several derivatives. We will use them later in the integral comutations. These solutions have been already showed in [3] and [], but we will show them ste by ste. We will also use the identity (3) and its variation with two arameters: sin x + cos x q =, see [3]. We assume x (, π ). d dx cos x = d dx ( sin x) = cos x sin = cos x sin x ( sin x) = () x = sin x x cos x = tan cos x cos d dx tan x = d sin x dx cos x = cos x sin x cos x cos x = + tan x = sin x cos x tan cos x cos x = (3) d dx cos,q x = d dx ( sin,q x) q = q ( sin,q x) q q = q cos q,q x sin,q x sin,q x cos,q x = (4) From the results above we assume that the methods of comuting integrals with generalized trigonometric functions will be a bit different from those in the chart.

16 3. Basic substitutions Let >, n >. An equal sign with a star means comutation by Wolfram Mathematica. 3.. cos x dx = sin x [Holds for x (, π π ) by definition, then for x (, π ) by cosine roerty cos x = cos ( x) (cosine is even) and for R by π -eriodicity.] 3.. f(sin x) cos x dx z = sin x, dz = cos x dx = f(z) dz = F (sin x) [Where x ( π, π ), f : (, ) R, F (x) = f(x).] 3..a π f(sin x) cos x dx z = sin x, sin dz = cos x dx = = F () F () [Where x ( π, π ), f : (, ) R, F (x) = f(x).] 3..b 3..3 π π f(z) dz = sin x cos x dx z = sin x, [ ] z dz = cos x dx = z dz = = z = sin x, sin n x dx dz = cos x dx, cos x = ( sin x) = z n dz = ( z ) ( ), n+; n+ + ; z = zn+ F n + [Where x ( π, π ).] π 3..3a sin n x dx ( = sinn+ x n + F z = sin x, dz = cos x dx, cos x = ( sin x) =, n + ; n + + ; sin x z n ( z ) ( Γ dz = f(z) dz = ) ) ( nγ ( Γ ) n n+ ) 3. Further comutations 3.. = tan x dx = sin x z ( z ) sin [Where x (, π 3.. sin x cos x dz = ). See () and [].] tan x dx = tan x x [Where x (, π ). See (3) and [].] x dx z = cos x, dz = cos dz = z sin x sin x dx x dx, z dz = ln z = ln cos x =

17 3..3 = cos x sin z = cos x, x dx dz = cos z dz = z+ cos+ + = + ). See ().] [Where x (, π dz = z sin x x sin x dx x dx, = z sin x z sin x dz = 3.3 Functions with two arameters 3.3. = tan,q x dx = sin,q x z q z q sin q = ( q) cosq,q x [Where x (, π 3.3. ). See (4).] cos,q x sin,q sin,q x cos,q x dz = q x dx z = cos,q x, dz = q cos q,q x sin,q x dx, dz = q z q sin,q x dx = z dz = q q+ z = cos,q x, x dx dz = q cos q,q dz = q z q sin,q z q+ q x sin,q x dx = q z q dz = q q + = (q + ) cosq+,q x [Where x (, π ). See (4).] x dx, z q dz = qzq (q ) = = z sin,q x q z q sin,q x dz = 3

18 4 Numerical art For comuting values of sin x, cos x and sin,q x we used Matlab solver ode45. From Matlab documentation [9] we know that the solver ode45 is based on Runge-Kutta (4,5) formula, the Dormand-Prince air. 4. Runge-Kutta method Runge-Kutta methods have roerties similar to the Taylor exansion, but they do not require analytic derivation. General Runge-Kutta method can be described by following equations [8]: k = hf(x n, y n ) k = hf(x n + α h, y n + β k ) k 3 = hf(x n + α h, y n + β k + β k ). k j+ = hf(x n + α j h, y n + β j k + β j k β jj k j ) y n+ = y n + γ k + γ k γ j+ k j+, where y = f(x, y), y(x ) = y is the initial value roblem, h is ste size. Coefficients α j and β jj are usually given in the following form. Table : Dormand-Prince coefficients scheme α β α β β α β 3 β 3 β 33. α 6 β 6 β 6 β 63 β 64 β 65 β 66 γ γ γ 3 γ 4 γ 5 γ 6 γ 7 γ γ γ3 γ4 γ5 γ6 γ7 Coefficients γ j give the fifth-order accurate method, coefficients γ j give the fourth-order accurate method. Dormand s and Prince s [4] research brought three sets of coefficients α j and β jj. For ode45 following coefficients are used. 4

19 Table 3: Dormand-Prince coefficients Matlab ode solvers comarison There are other solvers for numerical comuting differential equations in Matlab. We will comare ode3, ode45, ode3 and ode3t. We will solve differential equation v = w sgn(w), w = ( ) v sgn(v), v() =, w() = and we will comare values of sin x for = 6 and x = π 6 and values of sin x for x = π. The exact solutions are sin 6 (π 6 ) = and sin(π) =. Matlab solutions follow in the tables below. We used format long with 5 digits after the decimal oint. Table 4: Matlab ode solutions for sin 6 (π 6 ) Solver sin 6 (π 6 ) Error ode ode3t ode ode

20 Table 5: Matlab ode solutions for sin(π) Solver sin(π) Error ode ode3t ode ode Difference between exact values and comuted values is quite small. The least recise solver is ode3t. The reason might be the fact, that ode3t is determined to solve stiff equations. Other used solvers, ode3, ode45 and ode3, are set to solve non-stiff equations. We will not discuss stiffness of the equations in this thesis. We also comared values of sin x for x = π 4. The exact solution is sin( π 4 ) =. Table 6: Matlab ode solutions for sin( π) 4 Solver sin( π) Error 4 ode ode3t ode ode According to the tables above we can note that the error of ode45 solutions is for each examle on the th digit after the decimal oint. In the first two examles, ode3 is the most recise solver, but in the third examle the most recise one is ode45. 6

21 5 Imlementation We will introduce codes which were used to visualize generalized trigonometric functions in this thesis. We used MATLAB R4a, version 8.3 on rocessor Intel(R) Core(TM)i3-3M GHz. By using ode45 we solved the initial value roblem (7). Using odeset function we set relative tolerance and absolute tolerance RelTol = X Y min( X, Y ) AbsTol = X Y to 6. By choosing the tolerances we control the errors: e(i) max(relt ol y(i), AbsT ol(i)), where e(i) is error at ste i and y(i) is current solution [9]. 5. Values and grah of sin x and cos x The code in Listing 3 comutes values of sin x on, for different values of. Functions (Listing 5) and (Listing 6) stand for the initial value roblem (7) with = 6 and =.. The code in Listing 4 is analogous to the code in Listing 3, but comutes values of cos x on,. By solving first order differential equations we obtain two columns. The first column Y (:, ) stands for sin x. As we have defined in Chater, cos x def = d sin dx x. The second column is Y (:, ), we will name it y. Then we obtain: cos x = sgn(y ) y. The final grahs we can see in Figure and Figure 3. 7

22 Listing 3: Values and grah of sin x % initial value roblem: % y'=sign(y) y ^(/(-)) % y'=-lambda*sgn(y) y ^(-) % y()=,y()= otions=odeset('reltol',e-6,'abstol',e-6); t=[,]; init_y=; init_y=; figure('name', 'Sine'); axis([ - ]); hold on; [T,Y]=ode45(@,t,[init_y init_y],otions); %function sets =6 lot(t,y(:,),'black-','linewidth',.),ylabel('sin_x'),xlabel('x'); %grah of sin_6(x) [T,Y]=ode45(@,t,[init_y init_y],otions); %function sets =. lot(t,y(:,),'black:','linewidth',.),ylabel('sin_x'),xlabel('x'); %grah of sin_.(x) x=linsace(,,); y=sin(x); lot(x,y,'black-.','linewidth',.),ylabel('sin_x'),xlabel('x'); %grah of sin(x) legend('sin_6x','sin_{.}x','sin x','location','northeast'); hold off; 8

23 Listing 4: Values and grah of cos x % initial value roblem: % y'=sign(y) y ^(/(-)) % y'=-lambda*sgn(y) y ^(-) % y()=,y()= otions=odeset('reltol',e-6,'abstol',e-6); t=[,]; init_y=; init_y=; figure('name','cosine'); axis([ - ]); hold on; [T,Y]=ode45(@,t,[init_y init_y],otions); y=y(:,); %root y, see initial value roblem c=(sign(y)).*(abs(y)).^(/(6-)); %c is equivalent to cosine, see y' in initial value roblem lot(t,c,'black-','linewidth',.),ylabel('cos_x'),xlabel('x'); %grah of cos_6(x) [T,Y]=ode45(@,t,[init_y init_y],otions); y=y(:,); c=(sign(y)).*(abs(y)).^(/(.-)); lot(t,c,'black:','linewidth',.),ylabel('cos_x'),xlabel('x'); %grah of cos_.(x) y=cos(x); lot(x,y,'black-.','linewidth',.),ylabel('cos_x'),xlabel('x'); %grah of cos(x) legend('cos_6x','cos_{.}x','cos x','location','northeast'); hold off; 9

24 Listing 5: Function f u n c t i o n dy = (t,y) %suort function for ode45 =6; lambda=-; dy=[(sign(y())).*(abs(y())).^(/(-)); -*lambda*(sign(y())).*(abs(y())).^(-)]; end Listing 6: Function f u n c t i o n dy = (t,y) %suort function for ode45 =.; lambda=-; dy=[(sign(y())).*(abs(y())).^(/(-)); -*lambda*(sign(y())).*(abs(y())).^(-)]; end

25 5. Values and grah of π,q We wanted to find values of π,q and lot the grah for several fixed values of and q. We know that sin,q x = for x = kπ,q, k Z. We used bisection method with accuracy 3. The boundaries s and t were set to and 4. The while method continues till the current accuracy (mis = t s ) is bigger than 3. The variable tend (current x) is set to s+t. The boundaries change according to value of sin,q x in current tend. In Listing 7, {,.5, 3, 3.5, 4}, q {,.5, 3, 3.5, 4, 4.5, 5, 5.5, 6}. In Listing 8, q {,.5, 3, 3.5, 4}, {,.5, 3, 3.5, 4, 4.5, 5, 5.5, 6}. Both codes use the same functions search, containing the bisection method, and q, the suort function for ode45. We chose the figures for only, q {, 3.5, 4}. The final grahs are in Figure 7 and Figure 8. Other (easier) method of lotting grah of π,q is to visualize it according to definition (). However, the method we used would be suitable if the definition was unknown. Listing 7: Values and grah of π,q with fixed g l o b a l ; g l o b a l q; M=zeros(5,9); k=; %k values are used for indexing rows of M f o r j=:.5:4 %for each is made a vector of i_(q) l=; %l values are used for indexing columns of M f o r i=:.5:6 =j; q=i; tend=search(,4); M(k,l)=tEnd; l=l+; end k=k+; end m=:.5:6; axis([ 6 3.]); hold on; %values on x-axis lot(m,m(,:),'black--o','linewidth',.); %grah of i_,q as a function of q, = hold on lot(m,m(4,:),'black-.o','linewidth',.); %grah of i_,q as a function of q, =3.5 hold on lot(m,m(5,:),'black:o'),xlabel('q'),ylabel('\i_{,q}','linewidth',.); %grah of i_,q as a function of q, =4 legend('=','=3.5','=4') hold off

26 Listing 8: Values and grah of π,q with fixed q g l o b a l ; g l o b a l q; M=zeros(5,9); k=; %k values are used for indexing rows of M f o r j=:.5:4 %for each q is made a vector of i_q() l=; %l values are used for indexing columns of M f o r i=:.5:6 =i; q=j; tend=search(,4); M(k,l)=tEnd; l=l+; end k=k+; end m=:.5:6; axis([ 6 3.]); hold on; %values on x-axis lot(m,m(,:),'black--o','linewidth',.); %grah of i_,q as a function of, q= hold on lot(m,m(4,:),'black-.o','linewidth',.); %grah of i_,q as a function of, q=3.5 hold on lot(m,m(5,:),'black:o'),xlabel(''),ylabel('\i_{,q}','linewidth',.); %grah of i_,q as a function of, q=4 legend('q=','q=3.5','q=4') hold off

27 Listing 9: Function search f u n c t i o n [tend] = search(s,t) %finds value of i_,q by bisection method tend=; acc=.; %accuracy mis=; %mistake otions = odeset('reltol',e-6,'abstol',e-6); while mis>acc tend=(s+t)/; [T,Y]=ode45(@q,[,tEnd],[,],otions); mis=abs(t-s)/; i f Y(length(Y),)< %Y(length(Y),) is a value of sin_,q in tend t=tend; e l s e s=tend; end end end Listing : Function q f u n c t i o n dy = q(t,y) %suort function for ode45 g l o b a l ; g l o b a l q; lambda=q*(-)/; dy=[(sign(y())).*(abs(y())).^(/(-)); -*lambda*(sign(y())).*(abs(y())).^(q-)]; end 3

28 6 Conclusion We exlained the terms sin x, cos x, sin,q x, π and π,q and showed them from different oints of view, esecially in grahs. We also comared roerties of generalized trigonometric functions with roerties of their basic forms and tried to aly well known methods of comuting integrals with trigonometric functions to comutations of integrals with generalized trigonometric functions. According to the results of Chater 3 we can say that the only rule, which can be alied on integrals with both functions, basic and generalized, is the rule for f(sin x) cos x dx (Table ). We also showed the imlementation of the roblem in Matlab and described it in the numerical context. This thesis might initiate further research of the intersection of generalized sine and cosine function for different values of. Other follow-u might be a discussion of stiffness of the initial value roblem. Finally, more extensive research of the ode solvers in this context might be also useful. 4

29 References [] BHAYO, B. A., SÁNDOR, J.: Inequalities connecting generalized trigonometric functions with their inverses. Issues of analysis. (3), no.. [] BUSHELL, P. J., EDMUNDS, D. E.: Remarks on generalized trigonometric functions. Rocky Mountain J. Math. 4 (), no., [3] ČEPIČKA, J.: Comarison of analytical and numerical results for the -Lalace equation. Proceedings of the 7 Conference on Variational and Toological Methods: Theory, Alications, Numerical Simulations, and Oen Problems. Northern Arizona University, Flagstaff, Arizona. May 3-7, 7. [4] DORMAND, J. R., PRINCE, P. J.: A family of embedded Runge-Kutta formulae. J. Com. Al. Math., Vol. 6, 98, [5] EDMUNDS, D. E., LANG, J.: Generalizing trigonometric functions from different oints of view. Progresses in Mathematics, Physics and Astronomy (Pokroky MFA), vol. 4, 9, (in Czech). [6] ELBERT, Á.: A Half-linear second order differential equation. Colloq. Math. Soc. János Bolyai 3 (979), [7] GRADSHTEYN, I. S., RYZHIK, I. M.: Table of Integrals, Series, and Products. Elsevier/Academic Press, Amsterdam, 7. [8] KUBÍČEK, M., DUBCOVÁ, M., JANOVSKÁ, D.: Numerické metody a algoritmy. VŠCHT Praha,. ISBN [9] MathWorks - Makers of MATLAB and Simulink. URL: <htt:// [cit ] [] PRUDNIKOV, A. P., BRYCHKOV, Y. A., MARICHEV, O. I.: Integrals and series. Vol.. Elementary functions. Translated from the Russian and with a reface by N. M. Queen. Gordon and Breach Science Publishers, New York, 986. ISBN: [] PRUDNIKOV, A. P., BRYCHKOV, Y. A., MARICHEV, O. I.: Integrals and series. Vol.. Secial functions. Translated from the Russian by N. M. Queen. Second edition. Gordon and Breach Science Publishers, New York, 988. ISBN: [] TAKEUCHI, S.: Generalized ellitic functions and their alication to a nonlinear eigenvalue roblem with -Lalacian. Journal of Mathematical Analysis and Alications, vol. 358, no., 4-35, January. [3] WEI, D., LIU, Y.: Some generalized trigonometric sine functions and their alications. Alied mathematical sciences. 6 (), no.,