MATHEMATICA – AN INTRODUCTION:

R.C. VermaPhysics DepartmentPunjabi UniversityPatiala – 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

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] =

21.8088 + 0.142048 x + 0.0676953 x

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

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.

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

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

In[3] := Plot Gammax, x, 0.1, 5

Out[3]=

1 2 3 4 50

5

10

15

20

In[30]:= epsilon 10^ 16;

PlotAbsGamma1 I y ^2 Pi y SinhPi y, y, 2, 3,PlotRange epsilon, epsilon

Out[31]=2 .2 2 .4 2 .6 2 .8 3 .0

1 . 10 16

5 . 10 17

5 . 10 17

1 . 10 16

In[4] := Plot Abs Zeta 1 2 I y, y, 0, 40

Out[4]=

10 20 30 40

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

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 Yl

m(theta, phi)

Cyclotomic[n, z] gives Cyclotomic Cyn(z)

In[5] := Plot BesselJ0, x, x, 0, 30

Out[5]=

5 10 15 20 25 30

0 .4

0 .2

0 .2

0 .4

0 .6

0 .8

1 .0

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

Examples: on Hermite functions

In[163]:= h = HermiteH[8,x]Out[163] =

2 4 6 81680 - 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 62822400 - 45158400 x + 225792000 x - 373309440 x + 8 10 12 277831680 x - 103219200 x + 19726336 x - 14 16 1835008 x + 65536 x

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 p8 n + 96 p384 p

In[167]:= Solve[ {f1==1, f2==10, f3== 100}, {m,n,p}]Out[167] = 239 15 25

{{m -> ----, n -> -(--), p -> --}} 24 8 96

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// ExpandAllOut[169] = 2 2 (-1 + 3 z ) t1 + z t + -------------- + 2 2 5 z (-1 + 3 z ) 3 (-2 z + ---------------) t 2 --------------------------- + 3

4+ O[t]

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

In[10]:= Plot LegendreP3, Cosy, y, 0, 2Pi

Out[10]=1 2 3 4 5 6

1 .0

0 .5

0 .5

1 .0

In[172]:= Do[ Print["Cy",n," = ", Cyclotomic[n,z]], {n,6} ] Print["Cy", 50, " = ", Cyclotomic[50,z]]

Out[172] =Cy1 = -1 + zCy2 = 1 + z 2Cy3 = 1 + z + z 2Cy4 = 1 + z 2 3 4Cy5 = 1 + z + z + z + z 2Cy6 = 1 - z + z 5 10 15 20Cy50 = 1 - z + z - z + z

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}

Integral Transforms

FourierTransform Expt^2 Sint, t, 1 Cosh Sinh Cosh 1

41 2 Sinh 1

41 2

2 2

FourierTransform1, t, 2 DiracDelta

LaplaceTransformt^4Sint, t, s24 1 10 s2 5 s41 s2 5InverseLaplaceTransform1 1 s, s, t t

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}

77. FullForm of Mathematica commands

In[179]:= a+2 I b //FullFormOut[179] = Plus[a, Times[Complex[0, 2], b]]

In[180]:= FullForm[a + 2*I*b]Out[180] = Plus[a, Times[Complex[0, 2], b]]

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.

In[12]:= ArrayPlotCellularAutomaton50, RandomInteger1, 250, 100Out[12]=

End of part VII