Mathematical knowledge in Mathematica

The system contains all known special mathematical functions from pure and applied mathematics and engineering. They play a major role in solving problems from practice or mathematics.

Example 1. Generating Legeandre polynomials of arbitrary agree n,f or instance, let n=3 . By changing 3 with another natural number you will recieve other polynomials. The graphics is shown for thoroughness.

p = LegendreQ[3, x] Plot[p, {x, -1, 1}]

2/3 - (5 x^2)/2 - 3/4 x (1 - (5 x^2)/3) Log[(1 + x)/(1 - x)]

[Graphics:HTMLFiles/index_4.gif]

⁃Graphics⁃

Example 2. Mathematica can easily calculate symbolically complex integrals by applying all possible mathematical rules and transformations.

∫x^(1/2) ArcTan[x] x

-(4 x^(1/2))/3 + 1/3 2^(1/2) ArcTan[(-2^(1/2) + 2 x^(1/2))/2^(1/2)] + 1/3 2^(1/2) ArcTan[(2^(1 ... ) ArcTan[x] - Log[-1 + 2^(1/2) x^(1/2) - x]/(3 2^(1/2)) + Log[1 + 2^(1/2) x^(1/2) + x]/(3 2^(1/2))

Example 3. This goes for indefined integrals. We can get the numeric value by recalculating the result from the previous action with the command % // N:

∫_0^∞Log[x] Exp[-x^3] x %//N

1/81 Gamma[-2/3] (6 EulerGamma + 3^(1/2) π + 9 Log[3])

RowBox[{-, 0.932281}]

∫_0^∞Sin[x^2] Exp[-x] x %//N

1/4 (-2 HypergeometricPFQ[{1}, {3/4, 5/4}, -1/64] + (2 π)^(1/2) (Cos[1/4] + Sin[1/4]))

0.270514

Example 4. Finite and infinite sums and products can be calculated.

Underoverscript[∑, k = 1, arg3] 1/k^6 %//N Underoverscript[∏, k = 1, arg3] 1/k^6 %//N

52107472322919827957/51219253009612800000

1.01734

1/46656

0.0000214335

Look in the Mathematica demo  Integrals for more examples.

Example 5. All types of ordinary and partial differential equations can be symbolically solved if possible. Near we solve the ordinary differential equation   y'' + y' + x y =0 with regards to  x  and obtain its general solution depended on two arbitrary constants C[1] and C[2]. Then by means of the build-in function Evaluate[ ] we calculate the value of the obtained solution y[x] at  x=1 and constants C[1]=0, C[2]=1.

DSolve[y ' '[x] + y '[x] + x y[x] == 0, y[x], x] RowBox[{res, =, RowBox[{Evaluate, [, RowBox[{ ... 62754;, 1.}], ,, RowBox[{C[1], , 0.}], ,, , RowBox[{C[2], , 1.}]}], }}]}], ]}]}]

{{y[x] ^(-x/2) AiryAi[-(-1)^(1/3) (1/4 - x)] C[1] + ^(-x/2) AiryBi[-(-1)^(1/3) (1/4 - x)] C[2]}}

RowBox[{{, RowBox[{{, RowBox[{RowBox[{y, [, 1., ]}], , RowBox[{RowBox[{0.445528, }], +, RowBox[{0.170458, , }]}]}], }}], }}]

Example 6. Below are more examples for special functions and transformations. Here the correstness of the numeric inequalities is tested: when it is correct, the system is responding with True, otherwise it is False.

Log[2] <Zeta[3] <2^(1/2)

True

The following example is an application of the various trigonometric formulae by the build-in function TrigReduce[ ]  

TrigReduce[Cos[x]^4]

1/8 (3 + 4 Cos[2 x] + Cos[4 x])

Example 7. Checking if two numbers are prime or not by the function PrimeQ[ ]:

PrimeQ[242] PrimeQ[77431]

False

True

Created by Mathematica  (October 6, 2007)