More Math Help with Albert

[OneEye]

In Relativity, (p. 45), Dr. Einstein says:


...the energy of a material point of mass m is no longer gievn by the well-known expression

m { v \over c^2 }

but by the expression

{ m c^2 } \over \sqrt { 1 - { v^2 \over c^2 } }

...If we develop the expression for the kinetic energy in the form of a series, we obtain

mc^2 + m { v^2 \over 2 } + { 3 \over 8 } m { v^4 \over c^2 } . . . .

How does one get from the second equation to the third? What is meant by "develop[ing] the expression... in the form of a series"?

Any help would be more than appreciated.

[HallsofIvy]

One way is to use a "Taylor's series" expansion.

If f is any function, analytic at x0 then
f(x)= f(x0)+ f '(x0)(x- x0)+ (f''(x0)/2)(x- x0)2+ (f'''(x0)/6)(x- x0)3+ ...
the general term is (f(n)(x0)/n!)(x-x0)n where f(n) is the nth derivative.

In particular, use the "McLaurin series" which is the Taylor's series with x0= 0 and f(x)= (1+ x2/c2)-1/2, set x= v, and then multiply by mc2.

[OneEye]

HoI - Thanks - I suspected something like this, but lacked the mathematical toolkit to know what exactly was meant.

Most helpful!