in
MuPAD deserves the full support of the Linux community, and if you use mathematics in any way, then MuPAD should find a home on your system.
Number Theory

There are a few basic number theory functions in the kernel; others are contained in the numlib library.

```>> isprime(997);
TRUE
>> Factor(2^67-1);
193707721 761838257287
>> nextprime(1000000);
1000003
>> powermod(9382471,322973,1298377);
880825
>> phi(nextprime(2^20)-1);
498400
```

Here phi is Euler's totient function returning the number of integers less than and relatively prime to its argument. These functions allow us to perform simple RSA encryption and decryption. Suppose we choose two primes and compute their product:

```>> p:=nextprime(5678);
5683
>> q:=nextprime(6789);
6791
>> N:=p*q;
38593253
```
Now we have to choose an integer e relatively prime to (p-1)*(q-1); a smaller prime will do; say e:=17.
```>> e:=17:
```
The values e and N are our “public key”. Now we find the d, the inverse of e modulo (p-1)*(q-1). This is very easily done using the convenient overloading of the reciprocal function:
```>> d:=1/e mod (p-1)*(q-1);
6808373
```
Suppose someone wishes to send us a message M < N; say
```>> M:=24367139;
```
They can encrypt it using our public key values:
```>> M1:=powermod(M,e,N);
18476508
```
We can now decrypt this using the value d (and N):
```>> powermod(M1,d,N);
24367139
```
This is indeed the value of the original message.

Symbolic Manipulation

We have seen a glimpse of MuPAD's symbolic abilities in the equation solving above. But MuPAD can do much more than this: all manner of algebraic simplification; rewriting in a different form; partial fractions; and so on.

```>> expand((x+2*y-3*z)^4);
4       4       4         3      3            3       3            3
x  + 16 y  + 81 z  + 32 x y  + 8 x  y - 108 x z  - 12 x  z - 216 y z  -
3              2          2         2           2  2       2  2
96 y  z + 216 x y z  - 144 x y  z - 72 x  y z + 24 x  y  + 54 x  z  +
2  2
216 y  z
>> Factor(%);
4
(x + 2 y - 3 z)
>> sum(1/(k*(k+2)*(k+4)),k=1..n);
2        3       4
310 n + 337 n  + 110 n  + 11 n
----------------------------------------
2        3       4
4800 n + 3360 n  + 960 n  + 96 n  + 2304
>> partfrac(%);
1           1           1           1
--------- - --------- - --------- + --------- + 11/96
8 (n + 3)   8 (n + 1)   8 (n + 2)   8 (n + 4)
>> normal(%);
2        3       4
310 n + 337 n  + 110 n  + 11 n
----------------------------------------
2        3       4
4800 n + 3360 n  + 960 n  + 96 n  + 2304
>> Factor(%);
2
n (n + 5) (55 n + 11 n  + 62)
----------------------------------
96 (n + 1) (n + 2) (n + 3) (n + 4)
>> sum(sin(k*x),k=1..n);
(I exp(-I x) - I exp(I x) + I exp(-I n x) - I exp(I n x) -
I exp(- I x - I n x) + I exp(I x + I n x)) /
4 - 2 exp(-I x) - 2 exp(I x)
>> rewrite(%,sincos);
(2 sin(x) + 2 sin(n x) + I cos(x + n x) - 2 sin(x + n x) - I cos(- x - n x)
) / 4 - 4 cos(x)
```
Calculus

MuPAD's calculus skills are excellent. It implements very powerful algorithms for differentiation, integration, and limits.

```>> diff(x^3,x);
2
3 x
>> diff(exp(exp(x)),x\$4);
2                       3
exp(x) exp(exp(x)) + 7 exp(x)  exp(exp(x)) + 6 exp(x)  exp(exp(x)) +
4
exp(x)  exp(exp(x))
```

The dollar operator, \$, is MuPAD's sequencing operator. As with most operators, it is overloaded; in the context of a derivative it is interpreted as a multiple derivative. We can of course also perform partial differentiation.

```>> int(sec(x),x);
ln(2 sin(x) + 2)   ln(2 - 2 sin(x))
---------------- - ----------------
2                  2
>> int(cos(x)^3, x=-PI/4..PI/3);
1/2      1/2
5 2      3 3
------ + ------
12       8
>> int(E^(-x^2),x=0..0.5);
/    1                 \
int| --------, x = 0..0.5 |
|        2             |
|       x              |
\ exp(1)               /
>> float(%);
0.461281006412792448755702936740453103083759088964291146680472565934983884\
2952938567126622486999424745
```
If we require only a numeric result, then we don't want to force MuPAD to attempt a symbolic or exact solution first. In such a case we may use the hold command, which returns the input unevaluated, but “holds” onto it for the purposes of later evaluation. Thus we may enter:
```>> hold(int(exp(-x^2),x=0..0.5));
```
followed by the float command.

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