is not generally true for complex numbers.
I got confused on the operation and representation of complex numbers. It is
quite basic, and please help me understand. My question is as follows:
(1) Since 1=exp(2*pi*i), it seems that, for any real number a, we may have
exp(a*i)=exp((2*pi*i)*(a/(2*pi)))=(exp(2*pi*i))^(a/(2*pi))
= 1^(a/(2*pi))
=1
(2) For any complex number z, since 1+ exp(2*pi*i), it seems that
z^(1/4)=(1*z)^(1/4)=((exp(2*pi*i))^(1/4)*z^(1/4)
= (exp((pi/2)*i))*z^(1/4)
= i*z^(1/4)
by the same token, since 1=exp(-2*pi*i), it seems that
z*(1/4)=(-i)*z^(1/4)
All those are confusingly wrong, and I wish to understand why. Many thanks for you kindness to help.
Hi rcl333,
You confusion is due to the fact that some of the identities we take for granted about exponentiation of real numbers do not hold for complex numbers. In particular, it is not true that . Try reading the section on powers of complex numbers on this web page,
Exponentiation - Wikipedia, the free encyclopedia
especially the sub-section titled "Failure of Power and Logarithm Identities". Then if you are still confused, let us know.
I am extremely grateful for your kind and clear explanation with the Wikipedia reference. Now I understand that this rule does not always work for complex numbers. Please help me understand also when can we use this rule? and when can not?, because in Wikipedia site with the subsection "roots of the unity", it used the expression that (exp(2*pi*i*(1/n)))^k=exp(2*pi*i*k/n) in terms of the rule (e^a)^b=e^(ab).
Many thanks again.
The reason that
is that, in this case, it doesn't matter which branch of the complex logarithm of is chosen. The "obvious" choice is , but is also a valid choice for any integer m. (This is because .)
But for any choice of the branch of the logarithm, we have
because km is an integer.
The reason this trick does not work in general for is that is not, in general, an integer.
I appreciate the help earlier on the exponent rules on complex numbers. Please forgive my naiveness to the following followup questions:
Since for complex numbers, there are many branches of solutions for fractional power, how would we define which branch to choice, for example, if we have a function F(z)=a^z where a is a positive number (or even positive integer, this kind of function is basic in complex analysis), it shouldn't take multiple values, and which branch defines this function?
In particular, even in the real number syatem, 1 and -1 all serve as the squared root of 1. What is the meaning of 1^(1/2)? Which value and when should be adopted?
Thanks.
Principal value - Wikipedia, the free encyclopedia
The meaning of for is the positive number y such that .
In the real numbers, the convention is that always denotes the positive root. The familiar identities like depend on this convention.
In the complex numbers, no such convention works well in all cases, so if you need a single value of one of the multi-valued "functions", you must pick a branch of the function for that purpose. You may need to pick different branches for different problems. Fundamentally, however, one must simply be always aware of the fact that there are multiple possibilities for values of functions like the exponential and logarithmic functions.