| Posted on Saturday, 21 June, 2003 - 10:06 pm: |
Is there any difference, in terms of definition, between ex and exp(x)?
| Posted on Sunday, 22 June, 2003 - 07:40 am: |
Nope, there is no difference at all - just two ways of writing the same thing!
| Posted on Sunday, 22 June, 2003 - 12:10 pm: |
By convention they denote the same thing, although strictly the former is a multi-valued function and the latter is single-valued (for complex numbers). For complex numbers define,
| exp(z)= |
¥ å n=0 | zn/n! |
For w not equal to 0 define wz=exp(zlog(w)) =exp[z(ln|w|+i(argw+2np))] (any integer n). So if w=e then ez=exp(z)exp(2npiz) has many values. Take n=0 for the convention. Ian
| Posted on Sunday, 22 June, 2003 - 01:59 pm: |
Thanks.
One question though: is the definition of wz purely arbitrary?
Desmond
| Posted on Sunday, 22 June, 2003 - 02:08 pm: |
eiq can be seen to be equal to cosq+isinq by considering the power series expansions of ex, cosx and sinx. However, this assumes that the expansion of ex is valid for complex numbers, which at A-level we are simply told is acceptable, but which I vaguely recall reading somewhere is simply a matter of definition, i.e. we define eiq as cosq+i sinq.
| Posted on Sunday, 22 June, 2003 - 09:12 pm: |
as Philip said,
-Alex Miller
| Posted on Sunday, 22 June, 2003 - 11:08 pm: |
The above is all true, but the definition of zw is altogether more subtle. We define zw by zw=ew(logz) where we are taking logarithms to the base e. There are several things we have to consider here: 1) What it means to take the natural logarithm of a complex number - if we write z=r eiq (where r is real and q is the argument of z), then we may write that logz=log(r eiq)=logr+logeiq = logr+iq, 2) Note how we have to be careful with this, because the above complex logarithm is multivalued (i.e. since e2ip=1, then we could add any multiple of 2p to logz). Hence we restrict the argument of z to be within some range of width 2p (i.e. encompassing all of the complex plane), but such that there is no overlap, i.e. we may let (for example) -p < q £ p Such a restriction of the argument of z is known as a branch cut (in fact, the above branch is known as the principal branch). Another possibility would be 0 £ q < 2p. The whole point of this is to prevent the function z® zw being multivalued (i.e. the function is well defined). However, the range of q which we choose is purely arbitrary; in fact, different branches define different functions. However, it is relatively easy to show that this definition coincides with the (obviously single value) zn for integer n (in this case, the branch cut, since a change in the argument by 2p does not change zn, as e2ip=1). Bill