why (0+0i)^{0} == (nan, nan) in c++

Question

take a look at the code blew:

#include <complex>
#include <iostream>

int main()
{
    std::cout << std::pow( std::complex<double>(0,0), std::complex<double>(0,0) ) << "\n";
    std::cout << std::pow( std::complex<double>(0,0), double(0) ) << "\n";

    return 0;
}

g++(4.8.1) gives an output of

(nan,nan)
(-nan,-nan)

while clang++(3.3) gives an out put of

(-nan,-nan)
(-nan,-nan)

But I am expecting (1.0, 0.0).

Can anyone give an explanation?

@Mr.Alien: It does, I don't see why he is expecting what he is expecting. — PlasmaHH, Jun 28 '13 at 14:49
`zero` to the `zero` is undefined: http://www.math.utah.edu/~pa/math/0to0.html — Shafik Yaghmour, Jun 28 '13 at 14:51
@Mr.Alien: I don't think he is asking why he is expecting 1.0,0.0 ... — PlasmaHH, Jun 28 '13 at 14:51
mathematically it should be (1, 0), but why here g++/clang++ gives (nan,nan) -- that is the question. — Feng Wang, Jun 28 '13 at 14:51

taocp · Answer 1 · 2013-06-28T14:58:43.243

6

According to std::pow documentation

Return value
base raised by power (exp or iexp).
Domain error occurs if base is 0 and exp is less than or equal to 0. NAN is returned in that case. [...]

In your code, you have both base with 0 and exp equal to 0 since the complex number 0 + 0 *i is still 0. So NaN seems expected.

By courtesy of @Fred Larson, and according to overloaded std::pow for std::complex

Computes complex x raised to a complex power y. The operation is defined as exp(y · log(x) ). A branch cut exists along the negative real axis. The result of pow(0, 0) is implementation-defined.

edited Jun 28 '13 at 14:58

answered Jun 28 '13 at 14:51

taocp

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1

Actually, I'm not sure that citation is relevant, since `pow()` is overridden for `std::complex`. But this one is: http://en.cppreference.com/w/cpp/numeric/complex/pow Of note, "The result of pow(0, 0) is implementation-defined." – Fred Larson Jun 28 '13 at 14:54
@FredLarson Which would explain visual studio giving (1,0)(1,0)? – Floris Velleman Jun 28 '13 at 14:54
@FlorisVelleman: Sounds like it could return anything it wants to. – Fred Larson Jun 28 '13 at 14:56
@FredLarson Thanks for pointing that out, I updated the post. – taocp Jun 28 '13 at 14:58
before posting here, I have read the c++ standard draft n3376, but did not find this; thanks a lot. – Feng Wang Jun 28 '13 at 15:01

score 4 · Answer 2 · edited May 23 '17 at 12:28

As Fred Larson correctly points out the documentation says:

The result of pow(0, 0) is implementation-defined.

Mathematically this makes sense since we have a contradictory situation where N^0 should always be 1 but 0^N should always be 0 for N > 0, so you should have no expectations mathematically as to the result of this either. This Wolfram Alpha forum posts goes into a bit more details.

The case where the imaginary portion of the complex number is not zero is more complex situation. If the x in x^y is real then it should also be undefined but if x has an imaginary component then it looks like it is no longer undefined.

in Python also ? did you check ? – Grijesh Chauhan Aug 19 '13 at 18:08 — Grijesh Chauhan, Aug 19 '13 at 18:08

why (0+0i)^{0} == (nan, nan) in c++

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