why the absolute value of e(j*a) is one? where j=complex notation and a can have any value?

Answer 1

I am assuming j = one of the square roots of -1, and that you mean e to the ja power.

You are taking a number to a complex power. Usually that is defined as:

e^(x+iy) = e^x(cos(y)+j*sin(y))

If you substitute 0+ja for x+iy, you get:

e^(ja) = e^0 (cos(ja)+j sin(ja))

The absolute value of this is:

|e^(ja)| = |e^0||cos(ja)+jsin(ja))|

= 1*(cos^2(ja)+sin^2(ja))

= 1*1 = 1,

since cos^2(v)+sin^2(v) = 1 for any v.

So that is why the absolute value of e^(ja) is 1.

Answer 2

es, you're right: any real raised to any pure imaginary power gives a
complex number with absolute value 1, so the mapping y -> e^(iy) takes
the real line onto the unit circle, wrapping it around with a period
of 2 pi. That's what Euler's formula means: raising e to an imaginary
power produces the complex number with that angle.

You may want to look at our FAQ on this,

Imaginary Exponents and Euler's Equation
http://mathforum.org/dr.math/faq/faq.euler.equation.html

When we raise a real to a complex power, the imaginary component of
the exponent rotates it while the real component dilates (not
translates) it. That is, the angle of the resulting number is
determined by the imaginary part of the exponent, while the absolute
value is determined by the real part. This is what gives Euler's
formula tremendous power, and in fact this is the answer to the often-
asked question, what good are complex numbers? They let us combine
dilation and rotation, or exponential and sinusoidal functions, into a
single operation.

Multiplication by a complex number, similarly, rotates by the angle of
the complex number (adding the two angles), and dilates by its
absolute value (multiplying the absolute values.)

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/

Answer 3

e(j*a) means e times j times a. Is this what you intended?

Or, did you mean e(j^a)? If so, then for any value of a the absolute value is not 1.

Most likely, you intended e^j*a
in this case use DeMoivre's theorem which states that fo all x:
e^jx = cos x + j sin x

Now if you insert pi into the equation and get:

e^(j*pi*x) then it equals cos (pi*x) + jsin (pi*x).
Then for all integral values of x cos (pi*x) = +1 or -1 and sin (pi*j) =0.

Therefore e^(j*pi*x) = plus or - 1 and its absolute value is 1.

I hope this is what you wre really asking about.

Answer 4

are you sure you aren't looking at the modulus of a complex number? modulus is equivalent to absolute value:

modulus of, Be(j*a) = modulus(B) = B (for positive B)

so above, modulus is just 1, since B=1 and modulus(1)=1

[

modulus e(j*a) = 1, since,

e(j*a)=e(j*a)=cos(a)+i*sin(a)
and modulus {e(j*a)} = modulus {cos(a)+i*sin(a)} = sqrt{cos^2(a)+sin^2(a)}=1

]