Is there a way to transform the functionexp(A+B+C),where exp ⁡ ( ⋅ ) is the exponential function, into a sumf(A)+f(B)+f(C)?

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Is there a way to transform the functionexp(A+B+C),where   exp  ⁡  (  ⋅  ) is the exponential function, into a sumf(A)+f(B)+f(C)?

hocelwsmjc · Accepted Answer

If I am understanding correctly you would like to know if there is a suitable function f such that       e          a      +      b      +      c        =  f  (  a  )  +  f  (  b  )  +  f  (  c  ) for all values of a,b,c.Notice such a function would satisfy   f  (  0  )  +  f  (  0  )  +  f  (  0  )  =      e    0    =  1. So   f  (  0  )  =      1    3  .From here we can determine the function uniquely, since we must have       e    x    =  f  (  x  )  +  f  (  0  )  +  f  (  0  )  =  f  (  x  )  +      2    3  So the function f needs to be   f  (  x  )  =      e    x    −      2    3  We can see easily this doesn&#039;t work since if it did we would have       e    3    =  f  (  1  )  +  f  (  1  )  +  f  (  1  )  =  3  (  e  −      1    3    )  =  3  e  −  1 which clearly is not true.

assupecoitteem81 · Accepted Answer

Shorter solution for generalized version which proves there are no three functions f,g,h so that   f  (  a  )  +  g  (  b  )  +  h  (  c  )  =      e    x    :Note   f  (  x  )  =      e    x    −  g  (  0  )  −  h  (  0  ). We can do the same to the rest and get all the functions are       e    x   minus a constant.This isn&#039;t possible since then   f  (  x  )  +  g  (  x  )  +  h  (  x  )  =  3      e    x    −  c which is a lot smaller than       e          3      x       for large values of x.

Is there a way to transform the function exp(A+B+C), where exp(*) is the exponential function, into a sum f(A)+f(B)+f(C)?

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