MATHEMATICA – AN INTRODUCTION: R.C. Verma Physics Department Punjabi University Patiala – 147 002

1. MATHEMATICA – AN INTRODUCTION: R.C. Verma Physics Department Punjabi University Patiala – 147 002 PART VII- ADDITIONAL FEATURES Curve fitting Trigonometric and Exponential Expressions Testing functions Special Functions Orthogonal Polynomials Combinatorics: Permutations Fourier Transform FullForm of Mathematica commands

2. 70. Curve fitting: For a numerical data, it is often desired to find a simple polynomial function that approximates it. Fit[ data, {1, x}, x] gives the best linear fit Fit[ data, {1, x, x^2}, x] gives the best quadratic fit In[123]:= numdata = Table[ Exp[x/4.0], {x, 5}] Out[123] = {1.28403, 1.64872, 2.117, 2.71828, 3.49034} In[124]:= Fit[numdata, {1, x, x^2}, x] Out[124] = 2 1.8088 + 0.142048 x + 0.0676953 x

3. Examples: In[125]:= table = Table[ n!, {n,1,15}] Out[125] = {1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000 } In[126]:= logtable = N[ Log[table]] Out[126] = {0, 0.693147, 1.79176, 3.17805, 4.78749, 6.57925, 8.52516, 10.6046, 12.8018, 15.1044, 17.5023, 19.9872, 22.5522, 25.1912, 27.8993} In[127]:= Fit[logtable, {1,x,x^2,x^3}, x] Out[127] = 2 3 -0.700245 + 0.454654 x + 0.144605 x - 0.00320604 x

4. 71. Dealing with Trigonometric and Exponential Expressions: Mathematica has built-in functions for transforming Trigonometric and Exponential expressions. TrigExpand[ expression] expands out trigonometric expression into sum of trigonometric terms. TrigToExp[ expression] coverts trigonometric expression to exponential form. ExpToTrig[ expression] coverts exponential expression to trigonometric form.

5. 72. Testing functions (Result: True/False) IntegerQ[n] tests whether n is an integer, OddQ[n] tests whether n is odd, EvenQ[n] tests whether n is even, PrimeQ[n] tests whether n is a prime, In[156]:= OddQ[3] Out[156]= True

6. 73. Special Functions Beta[a, b] gives Euler beta function B(a, b) Gamma[ x] gives Euler gamma function, gamma(x). Zeta[s] gives Riemann zeta function zeta(s) Erf[z] gives error function erf(z) Example: In[157]:= N[ Zeta[ 1/3 + 15 I], 20] Out[157] = 0.008303807201447016 + 0.789018414658470727 I

10. 74. Orthogonal Polynomials BesselJ[n, x] gives Bessel function of first kind Jn(x) . BesselY[n, x] gives Bessel function of second kind Yn(x). HermiteH[n, x] gives Hermite Polynomial Hn(x) LaguerreL[n, x] gives Laguerrel Polynomial Ln(x) LegendreP[n, x] gives Legendre Polynomial Pn(x) LegendreP[n, m, x] gives associated Legendre Polynomial Pmn(x) SphericalHarmonicY[l, m, theta, phi] gives spherical Harmonics Ylm(theta, phi) Cyclotomic[n, z] gives Cyclotomic Cyn(z)

12. Examples: on Bessel functions In[158]:= BesselJ[0,15.2] Out[158] = -0.0544208 In[159]:= FindRoot[ BesselJ[1,x], {x, 8.0}] Out[159] = {x -> 7.01559} In[160]:= FindRoot[ BesselY[2, x], {x, 20.0}] Out[160] = {x -> 18.0642} In[161]:= f[n_, x_]:= BesselJ[n, x] f[0, 1.75] Out[161] = 0.369033 In[162]:= BesselJ[0,1.75] Out[162] = 0.369033

13. Examples: on Hermite functions In[163]:= h = HermiteH[8,x] Out[163] = 2 4 6 8 1680 - 13440 x + 13440 x - 3584 x + 256 x In[164]:= Print[ D[h,x]] Out[164] = 3 5 7 -26880 x + 53760 x - 21504 x + 2048 x In[165]:= Expand[h^2] Out[165] = 2 4 6 2822400 - 45158400 x + 225792000 x - 373309440 x + 8 10 12 277831680 x - 103219200 x + 19726336 x - 14 16 1835008 x + 65536 x

14. In[166]:= f[x_]:= m HermiteH[0,x] + n HermiteH[2,x] + p HermiteH[4,x] f1=f[1] f2=f''[1] f3=f''''[1] Out[166]= m + 2 n - 20 p 8 n + 96 p 384 p In[167]:= Solve[ {f1==1, f2==10, f3== 100}, {m,n,p}] Out[167] = 239 15 25 {{m -> ----, n -> -(--), p -> --}} 24 8 96

15. In[168]:= legen=Series[(1- 2 z t + t^2)^(-1/2), {t,0,4}] 2 z (-3 + 5 z ) Out[168] =------------- 2 In[169]:=Coefficient[Normal[legen], t,2]//Simplify// ExpandAll Out[169] = 2 2 (-1 + 3 z ) t 1 + z t + -------------- + 2 2 5 z (-1 + 3 z ) 3 (-2 z + ---------------) t 2 --------------------------- + 3 4 + O[t]

16. In[170]:= LegendreP[3,z] Out[170] = 3 -3 z 5 z ---- + ---- 2 2 In[171]:= leg[n_, z_]:= D[(z^2-1)^n, {z, n}]/(2^n n!) Print[Simplify[leg[3,z]]] Out[171] = 3 -3 z + 5 z ----------- 2

18. In[172]:= Do[ Print["Cy",n," = ", Cyclotomic[n,z]], {n,6} ] Print["Cy", 50, " = ", Cyclotomic[50,z]] Out[172] = Cy1 = -1 + z Cy2 = 1 + z 2 Cy3 = 1 + z + z 2 Cy4 = 1 + z 2 3 4 Cy5 = 1 + z + z + z + z 2 Cy6 = 1 - z + z 5 10 15 20 Cy50 = 1 - z + z - z + z

19. 75. Combinatorics: Permutations In[173]:= Permutations[{a,b,c}] Out[173] = {{a, b, c},{a, c, b},{b, a, c},{b, c, a}, {c, a, b}, {c, b, a}} In[174]:= Flatten[%] Out[174]= {a, b, c, a, c, b, b, a, c, b, c, a, c, a, b, c, b, a}

20. Integral Transforms

21. 76. Fourier Transform In[175]:= <<Calculus`FourierTransform` FourierTransform[t^2 Sin[t], t,w] Out[175] = -I Pi DiracDelta''[1 - w] + I Pi DiracDelta''[1 + w] In[176]:= data = {1,1,1,1, -1, -1,-1,-1}; In[177]:= N[Fourier[data]] Out[177]={0.+ 0. I, 1.20711 + 0.5 I, 0. + 0. I, 0.207107+ 0.5 I, 0. + 0. I, 0.207107 - 0.5 I, 0. + 0. I, 1.20711 - 0.5 I}

22. 77. FullForm of Mathematica commands In[179]:= a+2 I b //FullForm Out[179] = Plus[a, Times[Complex[0, 2], b]] In[180]:= FullForm[a + 2*I*b] Out[180] = Plus[a, Times[Complex[0, 2], b]]

23. ArrayPlotCellular automata provide a convenient way to represent many kinds of systems in which the values of cells in an array are updated in discrete steps according to a local rule.

24. End of part VII