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Imaginary part of expression too difficult to calculate

Finding real and imaginary partsMathematica taking too long to calculate recursive functionComplex Plot with Imaginary Part encoded in colorGetting the real part of a expressionAbout Complex Numbers, Real part and Imaginary part (symbolic calculus)Bug in HypergeometricPFQRegularized?Equivalence of ComplexExpand and assuming real argumentsSummation of complex and complex conjugate - elimination of imaginary partHow to get the real part of a complex expressionRoots of an expression

2

$begingroup$

I am trying to calculate the imaginary part of a long expression. It's a long enough expression that Mathematica "hangs" when you run:

imFUN2 = ComplexExpand[Im[expression]];

Is there something I can do that can help speed things up?

Here is my full code:

expression = -((I Ωc (4 γa^4 + 16 Δd^4 - 48 Δd^3 Δp + 48 Δd^2 Δp^2 - 16 Δd Δp^3 + 4 I γa^3 

  (3 Δc + 6 Δd - 4 Δp - Δs) - 16 Δd^3 Δs + 32 Δd^2 Δp Δs - 16 Δd Δp^2 Δs - 4 Δd^2 Ωc^2 + 

  4 Δd Δp Ωc^2 + 4 Δd Δs Ωc^2 - 4 I dephasing Δd Ωd^2 + 12 Δd^2 Ωd^2 + 

  4 I dephasing Δp Ωd^2 - 24 Δd Δp Ωd^2 + 12 Δp^2 Ωd^2 - 8 Δd Δs Ωd^2 + 

  8 Δp Δs Ωd^2 - Ωc^2 Ωd^2 + Ωd^4 + 4 Δc^2 (4 Δd^2 - 4 Δd Δp + Ωd^2) - 

  2 γa^2 (4 Δc^2 + 26 Δd^2 + 10 Δp^2 + 2 Δc (13 Δd - 7 Δp - 2 Δs) + 

  6 Δp Δs - 2 Δd (18 Δp + 5 Δs) - Ωc^2 + 4 Ωd^2) + 4 Δc (8 Δd^3 - 4 Δd^2 (4 Δp + Δs) - 

  (4 Δp + Δs) Ωd^2 + Δd (8 Δp^2 + 4 Δp Δs - Ωc^2 + 4 Ωd^2)) - 2 I γa (24 Δd^3 + 

  4 Δc^2 (3 Δd - Δp) - 4 Δp^3 - 4 Δp^2 Δs - 4 Δd^2 (13 Δp + 4 Δs) + Δp Ωc^2 + Δs Ωc^2 - 

  2 I dephasing Ωd^2 - 10 Δp Ωd^2 - 3 Δs Ωd^2 + Δc (36 Δd^2 + 8 Δp^2 + 4 Δp Δs - 

  4 Δd (11 Δp + 3 Δs) - Ωc^2 + 7 Ωd^2) + Δd (32 Δp^2 + 20 Δp Δs - 3 Ωc^2 + 10 Ωd^2))))/

  ((γa + 2 I Δd) (2 γa^2 - 4 Δc^2 + 4 Δd Δp - 4 Δp^2 + 4 Δd Δs - 8 Δp Δs - 4 Δs^2 + 

  2 I γa (3 Δc + Δd - 3 (Δp + Δs)) + Δc (-4 Δd + 8 (Δp + Δs)) + Ωd^2) (4 I γa^3 (Δc - Δp) - 

  16 Δc^2 Δd Δp - 16 Δc Δd^2 Δp + 16 Δc^2 Δp^2 + 48 Δc Δd Δp^2 + 16 Δd^2 Δp^2 - 32 Δc Δp^3 -

  32 Δd Δp^3 + 16 Δp^4 - 4 Δc Δd Ωc^2 - 4 Δd^2 Ωc^2 + 8 Δc Δp Ωc^2 + 8 Δd Δp Ωc^2 - 

  8 Δp^2 Ωc^2 + Ωc^4 - 4 Δc^2 Ωd^2 - 4 Δc Δd Ωd^2 + 8 Δc Δp Ωd^2 + 8 Δd Δp Ωd^2 - 

  8 Δp^2 Ωd^2 - 2 Ωc^2 Ωd^2 + Ωd^4 + 2 γa^2 (-4 Δc^2 - 6 Δc Δd + 14 Δc Δp + 6 Δd Δp - 

  10 Δp^2 + Ωc^2 + Ωd^2) - 2 I γa (4 Δc^2 (Δd - 3 Δp) - 4 Δd^2 Δp + Δd (20 Δp^2 - 

  3 Ωc^2 - Ωd^2) + Δc (4 Δd^2 - 24 Δd Δp + 28 Δp^2 - 3 Ωc^2 - Ωd^2) + 4 Δp (-4 Δp^2 + 

  Ωc^2 + Ωd^2)) + 2 dephasing (2 γa^3 + 2 I γa^2 (2 Δc + 3 Δd - 5 Δp) + γa (-4 Δd^2 - 

  4 Δc (Δd - 3 Δp) + 20 Δd Δp - 16 Δp^2 + Ωc^2 + Ωd^2) + 2 I (4 Δd^2 Δp - 8 Δd Δp^2 +

  4 Δp^3 - Δp Ωc^2 + Δd Ωd^2 - Δp Ωd^2 + Δc (4 Δd Δp - 4 Δp^2 + Ωd^2)))))) /.

      {γa -> 1, dephasing -> 10^-4};



imFUN2 = ComplexExpand[Im[expression]];

performance-tuning simplifying-expressions complex

share|improve this question

edited 3 hours ago

MarcoB

37.5k556113

asked 4 hours ago

Steven SagonaSteven Sagona

1866

$endgroup$

add a comment | 

2

$begingroup$

I am trying to calculate the imaginary part of a long expression. It's a long enough expression that Mathematica "hangs" when you run:

imFUN2 = ComplexExpand[Im[expression]];

Is there something I can do that can help speed things up?

Here is my full code:

expression = -((I Ωc (4 γa^4 + 16 Δd^4 - 48 Δd^3 Δp + 48 Δd^2 Δp^2 - 16 Δd Δp^3 + 4 I γa^3 

  (3 Δc + 6 Δd - 4 Δp - Δs) - 16 Δd^3 Δs + 32 Δd^2 Δp Δs - 16 Δd Δp^2 Δs - 4 Δd^2 Ωc^2 + 

  4 Δd Δp Ωc^2 + 4 Δd Δs Ωc^2 - 4 I dephasing Δd Ωd^2 + 12 Δd^2 Ωd^2 + 

  4 I dephasing Δp Ωd^2 - 24 Δd Δp Ωd^2 + 12 Δp^2 Ωd^2 - 8 Δd Δs Ωd^2 + 

  8 Δp Δs Ωd^2 - Ωc^2 Ωd^2 + Ωd^4 + 4 Δc^2 (4 Δd^2 - 4 Δd Δp + Ωd^2) - 

  2 γa^2 (4 Δc^2 + 26 Δd^2 + 10 Δp^2 + 2 Δc (13 Δd - 7 Δp - 2 Δs) + 

  6 Δp Δs - 2 Δd (18 Δp + 5 Δs) - Ωc^2 + 4 Ωd^2) + 4 Δc (8 Δd^3 - 4 Δd^2 (4 Δp + Δs) - 

  (4 Δp + Δs) Ωd^2 + Δd (8 Δp^2 + 4 Δp Δs - Ωc^2 + 4 Ωd^2)) - 2 I γa (24 Δd^3 + 

  4 Δc^2 (3 Δd - Δp) - 4 Δp^3 - 4 Δp^2 Δs - 4 Δd^2 (13 Δp + 4 Δs) + Δp Ωc^2 + Δs Ωc^2 - 

  2 I dephasing Ωd^2 - 10 Δp Ωd^2 - 3 Δs Ωd^2 + Δc (36 Δd^2 + 8 Δp^2 + 4 Δp Δs - 

  4 Δd (11 Δp + 3 Δs) - Ωc^2 + 7 Ωd^2) + Δd (32 Δp^2 + 20 Δp Δs - 3 Ωc^2 + 10 Ωd^2))))/

  ((γa + 2 I Δd) (2 γa^2 - 4 Δc^2 + 4 Δd Δp - 4 Δp^2 + 4 Δd Δs - 8 Δp Δs - 4 Δs^2 + 

  2 I γa (3 Δc + Δd - 3 (Δp + Δs)) + Δc (-4 Δd + 8 (Δp + Δs)) + Ωd^2) (4 I γa^3 (Δc - Δp) - 

  16 Δc^2 Δd Δp - 16 Δc Δd^2 Δp + 16 Δc^2 Δp^2 + 48 Δc Δd Δp^2 + 16 Δd^2 Δp^2 - 32 Δc Δp^3 -

  32 Δd Δp^3 + 16 Δp^4 - 4 Δc Δd Ωc^2 - 4 Δd^2 Ωc^2 + 8 Δc Δp Ωc^2 + 8 Δd Δp Ωc^2 - 

  8 Δp^2 Ωc^2 + Ωc^4 - 4 Δc^2 Ωd^2 - 4 Δc Δd Ωd^2 + 8 Δc Δp Ωd^2 + 8 Δd Δp Ωd^2 - 

  8 Δp^2 Ωd^2 - 2 Ωc^2 Ωd^2 + Ωd^4 + 2 γa^2 (-4 Δc^2 - 6 Δc Δd + 14 Δc Δp + 6 Δd Δp - 

  10 Δp^2 + Ωc^2 + Ωd^2) - 2 I γa (4 Δc^2 (Δd - 3 Δp) - 4 Δd^2 Δp + Δd (20 Δp^2 - 

  3 Ωc^2 - Ωd^2) + Δc (4 Δd^2 - 24 Δd Δp + 28 Δp^2 - 3 Ωc^2 - Ωd^2) + 4 Δp (-4 Δp^2 + 

  Ωc^2 + Ωd^2)) + 2 dephasing (2 γa^3 + 2 I γa^2 (2 Δc + 3 Δd - 5 Δp) + γa (-4 Δd^2 - 

  4 Δc (Δd - 3 Δp) + 20 Δd Δp - 16 Δp^2 + Ωc^2 + Ωd^2) + 2 I (4 Δd^2 Δp - 8 Δd Δp^2 +

  4 Δp^3 - Δp Ωc^2 + Δd Ωd^2 - Δp Ωd^2 + Δc (4 Δd Δp - 4 Δp^2 + Ωd^2)))))) /.

      {γa -> 1, dephasing -> 10^-4};



imFUN2 = ComplexExpand[Im[expression]];

performance-tuning simplifying-expressions complex

share|improve this question

edited 3 hours ago

MarcoB

37.5k556113

asked 4 hours ago

Steven SagonaSteven Sagona

1866

$endgroup$

add a comment | 

$begingroup$

I am trying to calculate the imaginary part of a long expression. It's a long enough expression that Mathematica "hangs" when you run:

imFUN2 = ComplexExpand[Im[expression]];

Is there something I can do that can help speed things up?

Here is my full code:

expression = -((I Ωc (4 γa^4 + 16 Δd^4 - 48 Δd^3 Δp + 48 Δd^2 Δp^2 - 16 Δd Δp^3 + 4 I γa^3 

  (3 Δc + 6 Δd - 4 Δp - Δs) - 16 Δd^3 Δs + 32 Δd^2 Δp Δs - 16 Δd Δp^2 Δs - 4 Δd^2 Ωc^2 + 

  4 Δd Δp Ωc^2 + 4 Δd Δs Ωc^2 - 4 I dephasing Δd Ωd^2 + 12 Δd^2 Ωd^2 + 

  4 I dephasing Δp Ωd^2 - 24 Δd Δp Ωd^2 + 12 Δp^2 Ωd^2 - 8 Δd Δs Ωd^2 + 

  8 Δp Δs Ωd^2 - Ωc^2 Ωd^2 + Ωd^4 + 4 Δc^2 (4 Δd^2 - 4 Δd Δp + Ωd^2) - 

  2 γa^2 (4 Δc^2 + 26 Δd^2 + 10 Δp^2 + 2 Δc (13 Δd - 7 Δp - 2 Δs) + 

  6 Δp Δs - 2 Δd (18 Δp + 5 Δs) - Ωc^2 + 4 Ωd^2) + 4 Δc (8 Δd^3 - 4 Δd^2 (4 Δp + Δs) - 

  (4 Δp + Δs) Ωd^2 + Δd (8 Δp^2 + 4 Δp Δs - Ωc^2 + 4 Ωd^2)) - 2 I γa (24 Δd^3 + 

  4 Δc^2 (3 Δd - Δp) - 4 Δp^3 - 4 Δp^2 Δs - 4 Δd^2 (13 Δp + 4 Δs) + Δp Ωc^2 + Δs Ωc^2 - 

  2 I dephasing Ωd^2 - 10 Δp Ωd^2 - 3 Δs Ωd^2 + Δc (36 Δd^2 + 8 Δp^2 + 4 Δp Δs - 

  4 Δd (11 Δp + 3 Δs) - Ωc^2 + 7 Ωd^2) + Δd (32 Δp^2 + 20 Δp Δs - 3 Ωc^2 + 10 Ωd^2))))/

  ((γa + 2 I Δd) (2 γa^2 - 4 Δc^2 + 4 Δd Δp - 4 Δp^2 + 4 Δd Δs - 8 Δp Δs - 4 Δs^2 + 

  2 I γa (3 Δc + Δd - 3 (Δp + Δs)) + Δc (-4 Δd + 8 (Δp + Δs)) + Ωd^2) (4 I γa^3 (Δc - Δp) - 

  16 Δc^2 Δd Δp - 16 Δc Δd^2 Δp + 16 Δc^2 Δp^2 + 48 Δc Δd Δp^2 + 16 Δd^2 Δp^2 - 32 Δc Δp^3 -

  32 Δd Δp^3 + 16 Δp^4 - 4 Δc Δd Ωc^2 - 4 Δd^2 Ωc^2 + 8 Δc Δp Ωc^2 + 8 Δd Δp Ωc^2 - 

  8 Δp^2 Ωc^2 + Ωc^4 - 4 Δc^2 Ωd^2 - 4 Δc Δd Ωd^2 + 8 Δc Δp Ωd^2 + 8 Δd Δp Ωd^2 - 

  8 Δp^2 Ωd^2 - 2 Ωc^2 Ωd^2 + Ωd^4 + 2 γa^2 (-4 Δc^2 - 6 Δc Δd + 14 Δc Δp + 6 Δd Δp - 

  10 Δp^2 + Ωc^2 + Ωd^2) - 2 I γa (4 Δc^2 (Δd - 3 Δp) - 4 Δd^2 Δp + Δd (20 Δp^2 - 

  3 Ωc^2 - Ωd^2) + Δc (4 Δd^2 - 24 Δd Δp + 28 Δp^2 - 3 Ωc^2 - Ωd^2) + 4 Δp (-4 Δp^2 + 

  Ωc^2 + Ωd^2)) + 2 dephasing (2 γa^3 + 2 I γa^2 (2 Δc + 3 Δd - 5 Δp) + γa (-4 Δd^2 - 

  4 Δc (Δd - 3 Δp) + 20 Δd Δp - 16 Δp^2 + Ωc^2 + Ωd^2) + 2 I (4 Δd^2 Δp - 8 Δd Δp^2 +

  4 Δp^3 - Δp Ωc^2 + Δd Ωd^2 - Δp Ωd^2 + Δc (4 Δd Δp - 4 Δp^2 + Ωd^2)))))) /.

      {γa -> 1, dephasing -> 10^-4};



imFUN2 = ComplexExpand[Im[expression]];

performance-tuning simplifying-expressions complex

share|improve this question

edited 3 hours ago

MarcoB

37.5k556113

asked 4 hours ago

Steven SagonaSteven Sagona

1866

$endgroup$

imFUN2 = ComplexExpand[Im[expression]];

expression = -((I Ωc (4 γa^4 + 16 Δd^4 - 48 Δd^3 Δp + 48 Δd^2 Δp^2 - 16 Δd Δp^3 + 4 I γa^3 

  (3 Δc + 6 Δd - 4 Δp - Δs) - 16 Δd^3 Δs + 32 Δd^2 Δp Δs - 16 Δd Δp^2 Δs - 4 Δd^2 Ωc^2 + 

  4 Δd Δp Ωc^2 + 4 Δd Δs Ωc^2 - 4 I dephasing Δd Ωd^2 + 12 Δd^2 Ωd^2 + 

  4 I dephasing Δp Ωd^2 - 24 Δd Δp Ωd^2 + 12 Δp^2 Ωd^2 - 8 Δd Δs Ωd^2 + 

  8 Δp Δs Ωd^2 - Ωc^2 Ωd^2 + Ωd^4 + 4 Δc^2 (4 Δd^2 - 4 Δd Δp + Ωd^2) - 

  2 γa^2 (4 Δc^2 + 26 Δd^2 + 10 Δp^2 + 2 Δc (13 Δd - 7 Δp - 2 Δs) + 

  6 Δp Δs - 2 Δd (18 Δp + 5 Δs) - Ωc^2 + 4 Ωd^2) + 4 Δc (8 Δd^3 - 4 Δd^2 (4 Δp + Δs) - 

  (4 Δp + Δs) Ωd^2 + Δd (8 Δp^2 + 4 Δp Δs - Ωc^2 + 4 Ωd^2)) - 2 I γa (24 Δd^3 + 

  4 Δc^2 (3 Δd - Δp) - 4 Δp^3 - 4 Δp^2 Δs - 4 Δd^2 (13 Δp + 4 Δs) + Δp Ωc^2 + Δs Ωc^2 - 

  2 I dephasing Ωd^2 - 10 Δp Ωd^2 - 3 Δs Ωd^2 + Δc (36 Δd^2 + 8 Δp^2 + 4 Δp Δs - 

  4 Δd (11 Δp + 3 Δs) - Ωc^2 + 7 Ωd^2) + Δd (32 Δp^2 + 20 Δp Δs - 3 Ωc^2 + 10 Ωd^2))))/

  ((γa + 2 I Δd) (2 γa^2 - 4 Δc^2 + 4 Δd Δp - 4 Δp^2 + 4 Δd Δs - 8 Δp Δs - 4 Δs^2 + 

  2 I γa (3 Δc + Δd - 3 (Δp + Δs)) + Δc (-4 Δd + 8 (Δp + Δs)) + Ωd^2) (4 I γa^3 (Δc - Δp) - 

  16 Δc^2 Δd Δp - 16 Δc Δd^2 Δp + 16 Δc^2 Δp^2 + 48 Δc Δd Δp^2 + 16 Δd^2 Δp^2 - 32 Δc Δp^3 -

  32 Δd Δp^3 + 16 Δp^4 - 4 Δc Δd Ωc^2 - 4 Δd^2 Ωc^2 + 8 Δc Δp Ωc^2 + 8 Δd Δp Ωc^2 - 

  8 Δp^2 Ωc^2 + Ωc^4 - 4 Δc^2 Ωd^2 - 4 Δc Δd Ωd^2 + 8 Δc Δp Ωd^2 + 8 Δd Δp Ωd^2 - 

  8 Δp^2 Ωd^2 - 2 Ωc^2 Ωd^2 + Ωd^4 + 2 γa^2 (-4 Δc^2 - 6 Δc Δd + 14 Δc Δp + 6 Δd Δp - 

  10 Δp^2 + Ωc^2 + Ωd^2) - 2 I γa (4 Δc^2 (Δd - 3 Δp) - 4 Δd^2 Δp + Δd (20 Δp^2 - 

  3 Ωc^2 - Ωd^2) + Δc (4 Δd^2 - 24 Δd Δp + 28 Δp^2 - 3 Ωc^2 - Ωd^2) + 4 Δp (-4 Δp^2 + 

  Ωc^2 + Ωd^2)) + 2 dephasing (2 γa^3 + 2 I γa^2 (2 Δc + 3 Δd - 5 Δp) + γa (-4 Δd^2 - 

  4 Δc (Δd - 3 Δp) + 20 Δd Δp - 16 Δp^2 + Ωc^2 + Ωd^2) + 2 I (4 Δd^2 Δp - 8 Δd Δp^2 +

  4 Δp^3 - Δp Ωc^2 + Δd Ωd^2 - Δp Ωd^2 + Δc (4 Δd Δp - 4 Δp^2 + Ωd^2)))))) /.

      {γa -> 1, dephasing -> 10^-4};



imFUN2 = ComplexExpand[Im[expression]];

share|improve this question

edited 3 hours ago

MarcoB

37.5k556113

asked 4 hours ago

Steven SagonaSteven Sagona

1866

share|improve this question

edited 3 hours ago

MarcoB

37.5k556113

asked 4 hours ago

Steven SagonaSteven Sagona

1866

edited 3 hours ago

MarcoB

37.5k556113

edited 3 hours ago

MarcoB

37.5k556113

asked 4 hours ago

Steven SagonaSteven Sagona

1866

asked 4 hours ago

Steven SagonaSteven Sagona

1866

1 Answer
1

active

oldest

votes

5

$begingroup$

Not an elegant solution but it works

num = Numerator[expression];

den = Denominator[expression];

{reNum, imNum} = ComplexExpand[ReIm[num]];

{reDen, imDen} = ComplexExpand[ReIm[den]];

(imNum reDen - reNum imDen)/(reDen^2 + imDen^2)

I think the problem is the denominator which is huge but I am surprised that my approach is not included in ComplexExpand

share|improve this answer

answered 3 hours ago

HughHugh

6,58421945

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This works for me, thanks. It is interesting though that you call ComplexExpand[ReIm[num]] inside your function. I guess I should always make sure I don't have fractions-inside-fractions before calling this. Also for completion it might be helpful for others to know that the real part is: (reNum reDen + imNum imDen)/(reDen^2 + imDen^2)
  $endgroup$
  – Steven Sagona
  3 hours ago
  

  
  
  
  

add a comment | 

Your Answer

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5

$begingroup$

Not an elegant solution but it works

num = Numerator[expression];

den = Denominator[expression];

{reNum, imNum} = ComplexExpand[ReIm[num]];

{reDen, imDen} = ComplexExpand[ReIm[den]];

(imNum reDen - reNum imDen)/(reDen^2 + imDen^2)

I think the problem is the denominator which is huge but I am surprised that my approach is not included in ComplexExpand

share|improve this answer

answered 3 hours ago

HughHugh

6,58421945

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This works for me, thanks. It is interesting though that you call ComplexExpand[ReIm[num]] inside your function. I guess I should always make sure I don't have fractions-inside-fractions before calling this. Also for completion it might be helpful for others to know that the real part is: (reNum reDen + imNum imDen)/(reDen^2 + imDen^2)
  $endgroup$
  – Steven Sagona
  3 hours ago
  

  
  
  
  

add a comment | 

5

$begingroup$

Not an elegant solution but it works

num = Numerator[expression];

den = Denominator[expression];

{reNum, imNum} = ComplexExpand[ReIm[num]];

{reDen, imDen} = ComplexExpand[ReIm[den]];

(imNum reDen - reNum imDen)/(reDen^2 + imDen^2)

I think the problem is the denominator which is huge but I am surprised that my approach is not included in ComplexExpand

share|improve this answer

answered 3 hours ago

HughHugh

6,58421945

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This works for me, thanks. It is interesting though that you call ComplexExpand[ReIm[num]] inside your function. I guess I should always make sure I don't have fractions-inside-fractions before calling this. Also for completion it might be helpful for others to know that the real part is: (reNum reDen + imNum imDen)/(reDen^2 + imDen^2)
  $endgroup$
  – Steven Sagona
  3 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Not an elegant solution but it works

num = Numerator[expression];

den = Denominator[expression];

{reNum, imNum} = ComplexExpand[ReIm[num]];

{reDen, imDen} = ComplexExpand[ReIm[den]];

(imNum reDen - reNum imDen)/(reDen^2 + imDen^2)

I think the problem is the denominator which is huge but I am surprised that my approach is not included in ComplexExpand

share|improve this answer

answered 3 hours ago

HughHugh

6,58421945

$endgroup$

num = Numerator[expression];

den = Denominator[expression];

{reNum, imNum} = ComplexExpand[ReIm[num]];

{reDen, imDen} = ComplexExpand[ReIm[den]];

(imNum reDen - reNum imDen)/(reDen^2 + imDen^2)

share|improve this answer

answered 3 hours ago

HughHugh

6,58421945

answered 3 hours ago

HughHugh

6,58421945

answered 3 hours ago

HughHugh

6,58421945