complex numbers in Symbolic Toolbox

Seems, I don't understand the meaning of imag and real in Symbolic Toolbox.

Suppose the following syms a b c d, F=(a+j*b)/(c+j*d). imag(F) gives -1/2*i*((a+i*b)/(c+i*d)-conj((a+i*b)/(c+i*d)))

This is obviously true, but practically for no use; I would like to see the result as

(bc-ad)/(c^2+d^2)

Or, another and simpler example:

F=a+j*b, imF=imag(F) gives -1/2*i*(a+i*b-conj(a+i*b)); The expressions for real(F) or imag(F) may not contain imaginary unit i. The expected answer would be imF=b.

Do you know a solution?

[Walter Roberson]

Opps! I deleted some of the text while reformatting. I have notified Juraj.

Answer 1

[Lucas García]

[Updated]

Define the symbolic variables as real and use the function simplify when needed:

>> syms a b c d real;
>> F=a+j*b; G=c+j*d;

imF=imag(F)

imF =

b

>> D = F/G;

imag(D) 
ans =

((a - b*i)*i)/(2*(c - d*i)) - ((a + b*i)*i)/(2*(c + d*i)) 

>> simplify(imag(D))

ans =

-(a*d - b*c)/(c^2 + d^2)

[Juraj]

Well, thanks, in this simple case it works.

However, for F=(a+j*b)/(c+j*d) it gives again the unusable answer

imag(F)=-1/2*i*((a+i*b)/(c+i*d)-conj((a+i*b)/(c+i*d)))

which is not acceptable, even if it is formally correct.

Juraj

[Lucas García]

Hi Juraj,

I have edited my previous response to include your comments.

[Juraj]

Hi, Lucas

as I see, it requires some experimentation.

Many thanks for inspiration.

Juraj

[Walter Roberson]

Also sometimes it helps to expand() and then simplify() again.

When working with imaginary values in Maple, sometimes I find I can make significant simplifications by simplify(combine(convert(EXPRESSION,expln))) and then converting back to real + i*imaginary format. Unfortunately I have not managed to find the MuPad equivalents of Maple's convert() or combine() functions.

[Walter Roberson]

From R2012a, rewrite() can be used to do some conversions; from R2014a, combine() can be used to combine operations.

Answer 2

[Modess Seyednezhad]

But I got this:

>> syms a b c d real; F=a+j*b; G=c+j*d; imF=imag(F)

imF =

b

>> D = F/G; >> k=imag(D)

k =

(b*c)/(c^2 + d^2) - (a*d)/(c^2 + d^2)

????

[Steven Lord]

And that expression for k is equivalent to the expression Juraj expected. Either simplify k or use numden to extract the numerator and denominator of k and it should match your expectation.