THE COMPLEX EXPONENTIAL FUNCTION

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Math307THECOMPLEXEXPONENTIALFUNCTION(The senotesassumeyouarealreadyfamiliarwithth ebasicpropertiesof complexnumbers.)We make thefollowingde nitionei = cos +isin :(1)Thisformulais calledEuler' orderto justifythisuseof theexponentialnotationappearingin (1), we will rstverifythefollowingformof theLaw ofExponents:ei 1+i 2=ei 1ei 2(2)To prove thiswe rstexpandtheright-handsideof (1)by rstmultiplyingouttheproduct:ei 1ei 2= (cos 1+isin 1)(cos 2+isin 2). Nextwe applyto thisthetrigonometricidentities:cos 1cos 2 sin 1sin 2= cos( 1+ 2)sin 1cos 2+ cos 1sin 2= sin( 1+ 2):Whenallthisis donetheresultisei 1ei 2= cos( 1+ 2) +isin( 1+ 2):Theright handsideof thelastequationis exactlywhatwe wouldgetif we wroteout(1)with replacedby 1+ 2. We have thereforeproved (2).To justifytheuseofe= 2:718: : : :, thebaseof thenaturallogarithm,in (1),we willdi erentiate(1)withrespectto : We shouldgetiei . Treatingilikeany otherconstant, we nddd ei =dd cos +isin = sin +icos.

THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following de nition ... we will rst verify the following form of the Law of Exponents: ei 1+i 2 = ei 1 ei 2 (2) To prove this we rst expand the right-hand side of (1) by rst multiplying out the

  Functions, Properties, Complex, Exponential, Exponent, Of exponents, The complex exponential function