## 1491 Reputation

8 years, 284 days

## FoxH function...

Looks like analytical solution for sum is FoxH function:

`sum(R^g*product(-B*r + N*g + 1, r = 1 .. g - 1)/(B^g*g!), g = 0 .. infinity) = -FoxH([[[1 + 1/B, 1 - N/B]], []], [[[0, 1]], [[1/B, -N/B]]], R)/B`

See attached file.

brn_ac_ver2.mw

## Analytical solution with Mathematica....

See attached file:

1_case1.mw

## Try:...

```INV := invztrans((z - 1)^2/(a*z^2 + b*z + c), z, n);
simplify(allvalues(INV));

#(-2^(-n)*((a - c)*sqrt(-4*a*c + b^2) + (b + 4*c)*a + c*b)*((-b + sqrt(-4*a*c + b^2))/a)^n + ((-a + c)*sqrt(-4*a*c + b^2) + (b + 4*c)*a + c*b)*(-(b + sqrt(-4*a*c + b^2))/(2*a))^n + 2*charfcn[0](n)*a*sqrt(-4*a*c + b^2))/(2*sqrt(-4*a*c + b^2)*a*c)```

## Try:...

Because dsolve doesn't know whether to compute  T(x) or T(L ) .

`dsolve({ics2, ode}, T(x));`

## Using Mathematica....

If we use MMA to solve series:

## One way is:...

```inttrans:-fourier(sech(x), x, -k);

#Pi*sech(k*Pi/2)```

## Try this:...

```L := 1;
g := 9.81;
f := 4*sqrt(L/g)*EllipticF(Pi/2, sin(theta0/2));
plot([Re(f), Im(f)], theta0 = 1 .. 20);```

With Maple 2022.2 and the option  adaptive I have:

 >
 >
 >

## Try...

```value(sol);
simplify(%);
```

I executed on Maple 2022.2 ,give me:

```#                  1      /               /
theta__1(y) = ---------- |27 (y + sigma) |
12 \               \
20 sigma
5                     9   /  10
- - lambda (k - 1) sigma  + |- -- lambda (k + 1) y
9                         \  27

/  149     149\  4   /49      49 \  3   /  29     29\  2
+ |- --- n + ---| k  + |--- n - ---| k  + |- -- n + --| k
\  432     432/      \108     108/      \  72     72/

/49      49 \     10            149     149\      8   11
+ |--- n - ---| k - -- Q lambda - --- n + ---| sigma  - -- (k
\108     108/     9             432     432/          6

/10  2          5          / 2   14      \
- 1) |-- y  lambda + -- (k + 1) |k  - -- k + 1| (n - 1) y
\33             36         \     11      /

/ 2   2       \\      7   ///  1     1\  4
+ Q (n - 1) |k  - -- k + 1|| sigma  + |||- - n + -| k
\     11      //          \\\  8     8/

/1     1\  3   /  3     3\  2   /1     1\     1
+ |- n - -| k  + |- - n + -| k  + |- n - -| k - - n
\2     2/      \  4     4/      \2     2/     8

10            1\  2
- -- Q lambda + -| y
9             8/

145                   / 2   50      \
- --- (k + 1) Q (n - 1) |k  - -- k + 1| y
108                   \     29      /

23  2 / 2   26      \        \      6                     /
- -- Q  |k  - -- k + 1| (n - 1)| sigma  - 4 (k - 1) (n - 1) |
6     \     23      /        /                            \
1                 2  3   1          2  2   5  2              3\
- -- (k + 1) (k - 1)  y  + - Q (k - 1)  y  + - Q  (k + 1) y + Q |
32                       4                 8                  /

5             /  1         4  4   3                  2  3
sigma  - 2 (n - 1) |- -- (k - 1)  y  - - Q (k + 1) (k - 1)  y
\  32               8

3  2        2  2   5  3              4\      4
+ - Q  (k - 1)  y  + - Q  (k + 1) y + Q | sigma  - 4 (k - 1) Q
2                  6                  /

/  1        2  2   3                2\  2      3
(n - 1) |- - (k - 1)  y  - - Q (k + 1) y + Q | y  sigma
\  8               8                 /

2          2 /  3        2  2   1                2\
- 2 Q  (n - 1) y  |- - (k - 1)  y  - - Q (k + 1) y + Q |
\  4               2                 /

2      3  4                          4  4        \
sigma  + 2 Q  y  (n - 1) (k - 1) sigma + Q  y  (n - 1)| (y
/

\
- sigma) Br|
/

```

Analytic_result_Help_ver2.mw

## Only: 1.77KiB :)...

Using:

```restart;
kernelopts(version);
#`Maple 2022.1, X86 64 WINDOWS, May 26 2022, Build ID 1619613`

A := factor(sum(sum(sum(sum(sum(sum(sum(sum(((x - i1)^2 + (y - i2)^2)*((x - j1)^2 + (y - j2)^2)*((x - k1)^2 + (y - k2)^2)*((x - l1)^2 + (y - l2)^2), i1 = 1 .. N), j1 = 1 .. N), k1 = 1 .. N), l1 = 1 .. N), i2 = 1 .. N), j2 = 1 .. N), k2 = 1 .. N), l2 = 1 .. N));

N := 8:
B := CodeTools:-Usage(A);

#memory used=1.77KiB, alloc change=0 bytes, cpu time=0ns, real time=0ns, gc time=0ns
#B := (5764801*(3*x^2 + 3*y^2 - 24*x - 24*y + 120)^4)/81```

## Alternative....

`plot([Re((-2)^x), Im((-2)^x)], x = -2 .. 2, legend = ["Real", "Imaginary"]);`

## Try:...

```kernelopts(version)
#`Maple 2022.1, X86 64 WINDOWS, May 26 2022, Build ID 1619613`

(int(sin(x)*exp(-2*I*Pi*f*x)/x, x = -infinity .. infinity) assuming (0 < f));

#signum(0, -2*Pi*f + 1, 0)*Pi/4 + Pi/2 - signum(0, 2*Pi*f - 1, 0)*Pi/4

convert(%, piecewise);

#piecewise(f < 1/(2*Pi), Pi, f = 1/(2*Pi), Pi/2, 1/(2*Pi) < f, 0)```

## Series solution:...

```Order := 50;
ode := diff(x(u), u, u) + 1/4*exp(-u^2)*x(u) = 0;
sol := convert(dsolve([ode, x(0) = 1, D(x)(0) = 0], x(u), 'series'), polynom);
plot([rhs(sol), diff(rhs(sol), u)], u = 0 .. 2, view = [0 .. 2, -1/2 .. 1])```

## 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));```

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