## 1491 Reputation

8 years, 223 days

## ....

If is indefine integral then is not possible.

integral.mw

## Only workaround....

INT := int(exp(-a^2-b^2-c^2), a = b^2/(4*c) .. infinity);

int(INT, b = -infinity .. infinity, c = 0 .. infinity, numeric);

#0.9786008283

Add option to roots: sqrt(2).

roots(x^3-x^2-8*x+8, sqrt(2));

#[[-2*sqrt(2), 1], [1, 1], [2*sqrt(2), 1]]

Thumb if You like.

## Use......

Use:

Student[Calculus1][DiffTutor]();

Enter a function: a(x)^b(x) a click AllSteps and Close then you see all steps.

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## Workaround....

At first read  this paper and everything will be clearer.

We can use (A.7 and A.9) and then:

Dirac(t) = Limit(exp(-t^2/epsilon^2)/(epsilon*sqrt(Pi)), epsilon = 0, right);

f := `assuming`([int((diff(exp(-t^2/epsilon^2)/(epsilon*sqrt(Pi)), t))*cos(t)*exp(-s*t), t = 0 .. infinity)], [s > 0, epsilon >= 0]);

limit(f, epsilon = 0, right);

# - infinity

Conclusion: Laplace((diff(Dirac(t), t))*cos(t), t, s) not exist in “generalized function” sense.

See attached file:

Laplace_Transfrom-Dirac-Heaviside-Derivative.mw

## Speed Up......

We can speed-up computation using dsolve, solving differential equation from definition: HeunC function.

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## No Bug...

It's not a bug it's so designed.

?sum,details

Use add than sum:

for i from 0 to 4 do add(1, j = i .. 1) end do;

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Simple code:

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## Works fine....

Work fine on Maple 2018.1. Probably you have older version of Maple,maybe try convert Matrix to 1D and then insert to procedure ?

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## Dot......

Try:

with(LinearAlgebra):

a := Matrix(2, 2, [[1, 2], [3, 4]]);

b := Matrix(2, 2, [[2, 4], [6, 2]]);

Multiply(a, b);

a.b

#Use a "dot" in keyboard

With this package "with(Student[LinearAlgebra])" works only if you enters: a.b

## Transcendental equation....

Try:

sol := solve(4^x+1 = x^4, {x}); evalf(allvalues(sol));

#{x = -1.053567011}, {x = 2.094012853}, {x = -.1688093539+1.061611481*I}, {x = -.1688093539-1.061611481*I}, {x = 3.989728952}

By numeric:

[fsolve(4^x+1 = x^4, x, -10 .. 0), fsolve(4^x+1 = x^4, x, 0 .. 3), fsolve(4^x+1 = x^4, x, 0 .. 10)];

#[-1.053567011, 2.094012853, 3.989728952]

Or:

f := a-> fsolve(4^x+1 = x^4, x = a); map(f, {-1, 2, 4});

#{-1.053567011, 2.094012853, 3.989728952}

# with inital starting values (-1,2,4)

[RootFinding:-Analytic(4^x+1-x^4, x, re = -5 .. 10, im = -2 .. 2)];

#[2.09401285285812, -1.05356701067272, 3.98972895158790, #-.168809353899946+1.06161148098136*I, -.168809353899951-1.06161148098136*I]

with(Student[NumericalAnalysis]): g :=a-> Newton(4^x+1-x^4, x = a, tolerance = 1/100000); map(g, {-1, 2, 4});

#{-1.053567011, 2.094012853, 3.989728951}

## Another workaround:...

```h := piecewise(abs(t) < 2*Pi*n, cos(t), 0);
h1 := convert(h, Heaviside);
inttrans:-fourier(h1, t, w) assuming n::posint;

#2*sin(2*Pi*n*w)*w/((w-1)*(w+1))```

## Forumlas for Prime Counting Function....

PRIMEPI := proc (x) options operator, arrow; evalf(add(`mod`(factorial(j-2), j), j = 4 .. x)) end proc;

PRIMEPI2 := proc (x) options operator, arrow; -1+add(factorial(j-2)-j*floor(factorial(j-2)/j), j = 3 .. x) end proc;

PRIMEPI3 := proc (x) options operator, arrow; add(sin(Pi*factorial(j-2)^2/j)^2/sin(Pi/j)^2, j = 2 .. x) end proc;

[PRIMEPI(29), PRIMEPI2(29), PRIMEPI3(29)];

#[10,10,10]

For large x code is very slow.

g := proc (x) options operator, arrow; numtheory[pi](floor(x)) end proc;#numtheory(for old Maple)

f := proc (x) options operator, arrow; NumberTheory:-PrimeCounting(x) end proc;

plot([PRIMEPI, g, f], 5 .. 100, color = ["Red", "Blue", "Green"], linestyle = [1, 4, 5]);

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