index 99a5a3c4e635273d3eb1dd05da15e760eb4d70a8..bf72d97320b667d1bcee62cf6572adee24845678 100644 (file)
@@ -695,51 +695,71 @@ ex power::conjugate() const

ex power::real_part() const
{

ex power::real_part() const
{
+       // basis == a+I*b, exponent == c+I*d
+       const ex a = basis.real_part();
+       const ex c = exponent.real_part();
+       if (basis.is_equal(a) && exponent.is_equal(c)) {
+               // Re(a^c)
+               return *this;
+       }
+
+       const ex b = basis.imag_part();
if (exponent.info(info_flags::integer)) {
if (exponent.info(info_flags::integer)) {
-               ex basis_real = basis.real_part();
-               if (basis_real == basis)
-                       return *this;
-               realsymbol a("a"),b("b");
-               ex result;
-               if (exponent.info(info_flags::posint))
-                       result = power(a+I*b,exponent);
-               else
-                       result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
-               result = result.expand();
-               result = result.real_part();
-               result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
+               // Re((a+I*b)^c)  w/  c ∈ ℤ
+               long N = ex_to<numeric>(c).to_long();
+               // Use real terms in Binomial expansion to construct
+               // Re(expand(power(a+I*b, N))).
+               long NN = N > 0 ? N : -N;
+               ex numer = N > 0 ? _ex1 : power(power(a,2) + power(b,2), NN);
+               ex result = 0;
+               for (long n = 0; n <= NN; n += 2) {
+                       ex term = binomial(NN, n) * power(a, NN-n) * power(b, n) / numer;
+                       if (n % 4 == 0) {
+                               result += term;  // sign: I^n w/ n == 4*m
+                       } else {
+                               result -= term;  // sign: I^n w/ n == 4*m+2
+                       }
+               }
return result;
}
return result;
}
-
-       ex a = basis.real_part();
-       ex b = basis.imag_part();
-       ex c = exponent.real_part();
-       ex d = exponent.imag_part();
+
+       // Re((a+I*b)^(c+I*d))
+       const ex d = exponent.imag_part();
return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
}

ex power::imag_part() const
{
return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
}

ex power::imag_part() const
{
+       const ex a = basis.real_part();
+       const ex c = exponent.real_part();
+       if (basis.is_equal(a) && exponent.is_equal(c)) {
+               // Im(a^c)
+               return 0;
+       }
+
+       const ex b = basis.imag_part();
if (exponent.info(info_flags::integer)) {
if (exponent.info(info_flags::integer)) {
-               ex basis_real = basis.real_part();
-               if (basis_real == basis)
-                       return 0;
-               realsymbol a("a"),b("b");
-               ex result;
-               if (exponent.info(info_flags::posint))
-                       result = power(a+I*b,exponent);
-               else
-                       result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
-               result = result.expand();
-               result = result.imag_part();
-               result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
+               // Im((a+I*b)^c)  w/  c ∈ ℤ
+               long N = ex_to<numeric>(c).to_long();
+               // Use imaginary terms in Binomial expansion to construct
+               // Im(expand(power(a+I*b, N))).
+               long p = N > 0 ? 1 : 3;  // modulus for positive sign
+               long NN = N > 0 ? N : -N;
+               ex numer = N > 0 ? _ex1 : power(power(a,2) + power(b,2), NN);
+               ex result = 0;
+               for (long n = 1; n <= NN; n += 2) {
+                       ex term = binomial(NN, n) * power(a, NN-n) * power(b, n) / numer;
+                       if (n % 4 == p) {
+                               result += term;  // sign: I^n w/ n == 4*m+p
+                       } else {
+                               result -= term;  // sign: I^n w/ n == 4*m+2+p
+                       }
+               }
return result;
}
return result;
}
-
-       ex a=basis.real_part();
-       ex b=basis.imag_part();
-       ex c=exponent.real_part();
-       ex d=exponent.imag_part();
+
+       // Im((a+I*b)^(c+I*d))
+       const ex d = exponent.imag_part();
return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
}

return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
}