## 1491 Reputation

8 years, 232 days

## MaplePrimes Activity

### These are answers submitted by Mariusz Iwaniuk

A := (a^6)^(1/3)*(-b^3)^(1/3)/a^3;

simplify(A) assuming a > 0, b < 0;

# -b/a

## Another way:...

Summary.:

For n=1,2,3,4.... plots are the same.

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## A way:...

If you have a list,then:

L := [a = b, e = f, g = w];

[seq(op(1, rhs(L[j])), j = 1 .. nops(L))];

#[b, f, w]

Or:

[seq(rhs(op(j, L)), j = 1 .. nops(L))];

## Successive approximations....

We can use  successive approximations. to solve integro-eqaution.

But in yours case is extremely very slow and  not very accuracy,you need many terms of series.

It's not a realy an answer,but  give you some insights about solution (How look like on plot).

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## Workaround....

```integral := Int(exp(-(2*m-4)*exp(t)+t*(m+1))*(t-2*exp(t)), t = 0 .. infinity);
with(IntegrationTools):
simplify(Change(integral, t = ln(1/x)));
```

then:

```f3 := proc (m) options operator, arrow; evalf(Int(-exp((-x*(m+1)*ln(x)-2*m+4)/x)*(ln(x)*x+2)/x^2, x = 0 .. 1, method = _d01ajc)) end proc;
plot(f3, 3 .. 20);```

## Series.....

Approximate only by infinity Series:

Int(exp(-1-1/v)*(1-exp(v^2/(2*t^2)))/v^2, v = 0 .. 1) = 1/exp(2)-Sum(2^(-j)*t^(-2*j)*GAMMA(-2*j+1, 1)*exp(-1)/factorial(j), j = 0 .. infinity);

1/exp(2.)-evalf[10](Sum(eval(2^(-j)*t^(-2*j)*GAMMA(-2*j+1, 1)*exp(-1)/factorial(j), t = 1), j = 0 .. infinity));

#-0.317747611e-1

int(eval(exp(-1-1/v)*(1-exp(v^2/(2*t^2)))/v^2, t = 1), v = 0 .. 1, numeric);

#-0.3177476104e-1

## Symbolic solution....

dsolve command this type equation can't solve.

In Maple exist another build-in command to solve  integral equation intsolve,but returns unevaluated for me(an error messages).

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 -0.195342e-92 -0.195343e-92 -0.195343e-92 -0.195348e-92 -0.195342e-92 -0.195292e-92 -0.195332e-92 -0.195232e-92 -0.194432e-92 -0.193432e-92
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## dsolve....

Use a dsolve build-in function to solve ODE's:

dsolve(odetemp = 0, U(z));

#U(z) = (1/2)*(2*tan(sqrt(_C1*c)*(_C2+z)*sqrt(2)/(2*c))*sqrt(_C1*c)+sqrt(2)*v)*sqrt(2)

If dsolve can't solve, the only hope is numeric integration the ODE's or another methods(e.g: series,...).

dsolve([odetemp = 0,ics,bcs], numeric);

maplerpimes_v2.mw

## ....

#It's a Bug in method = lookup, use another method like:"FTOC" or "FTOCMS"

int(exp(-t)/(1-t), t = 0 .. infinity, CauchyPrincipalValue = true, method = FTOC);

#Ei(1)*exp(-1)

`You can see what method uses the int,execute code:`
```restart:

infolevel[IntegrationTools] := 3;

int(exp(-t)/(1-t), t = 0 .. infinity, CauchyPrincipalValue = true);```

## Maple simply does not know how to  ...

Maple simply does not know how to  find closed form of the SUM.

Limit needs this solution from sum,but return unevaluated and for that  this limit evaluation is somehow broken.

Try this only approx:

```f := proc (x, n) options operator, arrow; evalf(x*ln(n)-(1/2)*Pi-add(arctan(x/k), k = 1 .. n)) end proc;
x := 2; INF := 10^6; [f(x, INF), evalf(argument(GAMMA(I*x)))];

#[-1.441148036, -1.441150010]```

EDITED 03.08.2018:

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## Weakness of Maple product....

Maple is weak solving inequalities and this case give a incorrect answer(probably a bug).

See attached file:

## Numerically....

You can compute numerically this integeral if you assume for constant p:

```p := 10;
KK := int(-87.20805639*csgn((p^2+16.703569+(10.0-q)^2+8.174*p)/sqrt((p^2+16.703569+(10.0-q)^2)^2-66.814276*p^2))*(2.*(p^2+16.703569+(10.0-q)^2)^2+66.814276*p^2)/(sqrt((p^2+16.703569+(10.0-q)^2)^2-66.814276*p^2)*((p^2+16.703569+(10.0-q)^2)^4-133.628552*(p^2+16.703569+(10.0-q)^2)^2*p^2+4464.147477*p^4)), q = 0 .. infinity, numeric);```

#KK := -0.5804485609e-2

Or Plotting:

```func := proc (p) options operator, arrow; evalf(Int((-1)*87.20805639*csgn((p^2+16.703569+(10.0-q)^2+8.174*p)/sqrt((p^2+16.703569+(10.0-q)^2)^2+(-1)*66.814276*p^2))*(2.*(p^2+16.703569+(10.0-q)^2)^2+66.814276*p^2)/(sqrt((p^2+16.703569+(10.0-q)^2)^2+(-1)*66.814276*p^2)*((p^2+16.703569+(10.0-q)^2)^4+(-1)*133.628552*(p^2+16.703569+(10.0-q)^2)^2*p^2+4464.147477*p^4)), q = 0 .. infinity, method = _d01amc)) end proc;
plot([seq([p, func(p)], p = -2 .. 2, 1/10)]);```

## combine...

```combine(log(x)+log(y)) assuming(x > 0, y > 0);
#ln(x*y)

combine(log(x)+log(y)) assuming(x > 0);
#ln(x*y)

combine(log(x)+log(y)) assuming(y > 0);
#ln(x*y)```

## Only hope is numeric......

restart;

de := sin(1)*(diff(y(x), x\$2))+(1+cos(1)*x^2)*y(x) = -1;

cond := y(-1) = 0, y(1) = 0;

sol := dsolve({cond, de}, numeric);

plots:-odeplot(sol, [x, y(x)]);

It is  not so easy.Only hope is numericbecause dsolve must find integral(probably closed form not exist)  and solve a Transcendental equation(Such equations often do not have closed form solution ) with WhittakerM function.

We can try to find a general solution.

value(dsolve(de));

but Maple can't find  integral and returns unevaluated.

## Factor 40......

Borrowing code from user https://www.mapleprimes.com/users/tomleslie and adding method _d01ajc to integrate(changing limits to [0,1]) we can speed up code at factor 40,and takes about 2.828 secs on my machine (see the attached).

sumInt-SPEED.mw

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