Euler Formula: e^(pi*i) = -1Date: 6/5/96 at 23:18:24 From: Lucas W Tolbert Subject: Euler Formula: e^(pi*i) = -1 Could you please explain why e^(pi*i) = -1? Date: 6/6/96 at 12:28:49 From: Doctor Anthony Subject: Re: Euler Formula: e^(pi*i) = -1 There is the well known identity that e^(ix) = cos(x) + i.sin(x) Then let x = pi. cos(pi) = -1, sin(pi) = 0 so e^(i.pi) = cos(pi) + i.sin(pi) = -1 + 0 and so e^(i.pi) = -1 From this you can also write the FAMOUS FIVE equation connecting the five most important numbers in mathematics, 0, 1, e, pi, i e^(i.pi) + 1 = 0 To show the truth of the identity quoted above we let z = cos(x) + i.sin(x) Then dz/dx = -sin(x)+i.cos(x) = i{cos(x)+i.sin(x)} = i.z So dz/z = i.dx Now integrate ln(z) = i.x + const. When x=0, z=1 so const=0 ln(z) = i.x z = e^(i.x) So cos(x) + i.sin(x) = e^(i.x) -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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