## 1511 Reputation

8 years, 338 days

## Workaround...

Looks like a bug in integrate.

`int(x^n*exp(x), x = 0 .. 2, method = _RETURNVERBOSE); #Can't solve.`

I have only a workround.

See attached file:2_ver1.mw

## Sum only....

As a infinite sum:

`Int(sin(sqrt(-x^2 + 1)), x) = Sum(2^(-1/2 - k)*sqrt(Pi)*x^(1 + 2*k)*BesselY(-1/2 + k, 1)/((1 + 2*k)*GAMMA(1 + k)), k = 0 .. infinity) + C`

for: -1<=x<=1

Check:

```int(sin(sqrt(-x^2 + 1)), x = 0 .. 1, numeric);
evalf(add(eval(2^(-1/2 - k)*sqrt(Pi)*x^(1 + 2*k)*BesselY(-1/2 + k, 1)/((1 + 2*k)*GAMMA(1 + k)), x = 1), k = 0 .. 2000));```

## Solve by rsolve...

 >

 (1)
 >
 (2)
 >
 (3)

 (4)

.

```f := (x, y) -> piecewise(x <> 0 and y <> 0, (y^2 + x^2)^x, x = 0 and y = 0, 1);
DX := diff(f(x, y), x);
DY := diff(f(x, y), y);
[eval(DX, [x = 1, y = 1]), eval(DY, [x = 1, y = 1])]

#at x=1 and y=1;
#[2*ln(2) + 2, 2]```

 (1)

 (2)

 (3)

 (4)

 (5)

## Another way....

```restart;
with(LinearAlgebra);
w := (2*Pi)/14;
v := Vector([1, sin(w*t), cos(w*t)]);
simplify(eval(sum(v . (Transpose(v)), t = k .. k + n), n = 13));```

## Try:...

```restart;
ODE := diff(W(x), x \$ 2) + (A*ohm^2*ro - kd)*W(x)/T = p;
sol1 := dsolve(ODE, W(x));
assign(sol1);

subs(_C1 = 0, W(x));

#or:

eval(W(x), _C1 = 0);

#sin(sqrt(A*ohm^2*ro - kd)*x/sqrt(T))*_C2 + p*T/(A*ohm^2*ro - kd)
```

## Another way:...

```int(exp(-2*r)*cos(theta)^3*r^2*sin(theta), phi = 0 .. 2*Pi, r = 0 .. infinity, theta = 0 .. Pi);
#O
```

## Series solution....

We can find solution with series representation:

```restart;
odeSystem := {diff(y1(x), x) = -x*y2(x) - (1 + x)*y3(x), diff(y2(x), x) = -x*y1(x) - (1 + x)*y4(x), diff(y3(x), x) = -x*y1(x) - (1 + x)*y4(x) - 5*x*cos(1/2*x^2), diff(y4(x), x) = -x*y2(x) - (1 + x)*y3(x) + 5*x*sin(1/2*x^2), y1(0) = 5, y2(0) = 1, y3(0) = -1, y4(0) = 0};
Order := 10;
systemSol := dsolve(odeSystem, [y1(x), y2(x), y3(x), y4(x)], series);
F := convert(systemSol, polynom);
plot([rhs(F[1]), rhs(F[2]), rhs(F[3]), rhs(F[4])], x = 0 .. 1, color = [red, blue, green, gold], legend = [y1(x), y2(x), y3(x), y4(x)]);```

## A workaround:...

A workaround with Mathematica:

```f[k_] := Sum[(-1)^i (k - i + 1)^(2 k + 4)/((i!)*((2 k - i + 2)!)), {i,0, k}];
L = Table[f[k], {k, 0, 1000}];
FindSequenceFunction[L, k]

(*1/12 (k + 3 k^2 + 2 k^3)*)```

With Maple:

```restart;
with(gfun):
f := k -> sum((-1)^i*(k - i + 1)^(2*k + 4)/(i!*(2*k - i + 2)!), i = 0 .. k);
l := [seq(f(k), k = 0 .. 1000)];
rec := listtorec(l, u(k), [ogf]);
rsolve(op(1, rec), u(k));

#1/6*k^3 + 3/4*k^2 + 13/12*k + 1/2```

## Workaround....

See attached file.

integral.mw

## Another way....

```Sum((-1)^k*sin(k*x)/k, k = -infinity .. infinity) = evalc(sum((-1)^k*sin(k*x)/k, k = -infinity .. infinity, formal))
```

## One way is:...

```f := x -> convert(sin(theta), x);
map(f, [cos, tan, cot, csc, sec]);```

#[-cos(theta + Pi/2), 2*tan(theta/2)/(1 + tan(theta/2)^2), 2*cot(theta/2)/(cot(theta/2)^2 + 1), 1/csc(theta), -1/sec(theta + Pi/2)]

## Numerically....

```sol := dsolve({diff(f(x), x) = f(1/x), f(0) = 0}, numeric, delaymax = 1.0, delaypts = 2000);
plots:-odeplot(sol, [x, f(x)], x = -1 .. 1);```

Maybe you can try with another initial condition f(0)=1 ?

## Simpler....

```evalf[30](subs(z = 2 + 3*I, Zeta(-z) + 2*z!*sin(Pi*z/2)*Zeta(z + 1)/(2*Pi)^(z + 1)));

rtable([seq(evalf[2^m](subs(z = 2 + 3*I, Zeta(-z) + 2*z!*sin(Pi*z/2)*Zeta(z + 1)/(2*Pi)^(z + 1))), m = 1 .. 10)], subtype = Vector[column]);#Depending of the precise```
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