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Applying "FullSimplify" fails to simplify the following expressions:

Input:

FullSimplify[(-0.0001+0.01*I)*E^((-1.6*10^7-2.1*10^8*I)*t)-(0.0001+0.01*I)*E^((-1.6*10^7+2.1*10^8*I)*t)]

Output:

(-0.0001 + 0.01 I) E^((-1.6*10^7 - 2.1*10^8 I) t) - (0.0001 + 0.01 I) E^((-1.6*10^7 + 2.1*10^8 I) t)

However, in fact, by applying Euler's formula, the above expression can be manually simplified to:

-2*(0.0001*Cos[-2.1*10^8*t] + 0.01*Sin[-2.1*10^8*t])*E^(-1.6*10^7*t)

How can I do to simplify to this form in Mathematica？

Thank you very much!

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2
  • $\begingroup$ Have you already seen ExpToTrig[]? $\endgroup$
    – J. M.'s eventual burnout ♦
    Jun 10, 2022 at 15:52
  • $\begingroup$ ExpToTrig[] makes the expression more complex. The output is: (-0.0001 - 0.01 I) Cos[(2.1*10^8 + 1.6*10^7 I) t] - (0.0001 - 0.01 I) Cosh[(1.6*10^7 + 2.1*10^8 I) t] + (0.01 - 0.0001 I) Sin[(2.1*10^8 + 1.6*10^7 I) t] + (0.0001 - 0.01 I) Sinh[(1.6*10^7 + 2.1*10^8 I) t]. Moreover, the terms of Sinh[] and Cosh[] cannot be cancelled out with TrigExpand[] or TrigReduce[]. $\endgroup$
    – Jeremy
    Jun 10, 2022 at 15:59

1 Answer 1

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expr = (-0.0001 + 0.01*I)*
   E^((-1.6*10^7 - 2.1*10^8*I)*t) - (0.0001 + 0.01*I)*
   E^((-1.6*10^7 + 2.1*10^8*I)*t)

ComplexExpand[expr, TargetFunctions -> {Abs, Arg}] // 
  TrigReduce // Chop

$$-0.0002 e^{-1.6\times 10^7 t} (1. \cos (2.1\times 10^8 t)-100. \sin (2.1\times 10^8 t))$$

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  • 1
    $\begingroup$ Thanks, your method solved my problem! $\endgroup$
    – Jeremy
    Jun 11, 2022 at 13:47

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