- #1
dave4000
- 16
- 0
Is this true:
[tex]e^{i(a+bx)}=cos(a+bx)+i sin(a+bx)[/tex] ?
Last edited by a moderator:
Confirming the Euler's Formula: e^{i(a+bx)}=
- Thread starter dave4000
- Start date
-
- Tags
- Formula
In summary: In general, this isn't true (in fact, for cos x = cos y, we have that x = y +/- 2k Pi for some integer k). This periodicity makes it impossible for the logarithm to be a true inverse function of the exponential function, as it would require a single-valued function, which the logarithm is not.In summary, the exponential function is not a bijection over the complex numbers, meaning that e^x = e^y does not imply that x = y. This is due to the periodicity of the complex exponential function. Therefore, the logarithmic function cannot be a true inverse function of the exponential function, as it would require a single-valued function. Instead, the logarith
Is this true:
[tex]e^{i(a+bx)}=cos(a+bx)+i sin(a+bx)[/tex] ?
- #2
chroot
Staff Emeritus
Science Advisor
Gold Member
- 10,296
- 41
Yes -- that's the beauty of function notation. You can replace 'x' with any expression you like.
- Warren
- Warren
- #3
dave4000
- 16
- 0
Thanks
- #4
Scandal
- 4
- 0
I know Daves question is answered, but instead of making a new thread I just ask another one here as it is somewhat related. Unfortunately I don't have a mathtype program so you'll have to imagine.
So if you have the e^xi expression and set x = pi, this will equal -1. Adding 2Pi equals 1. So far so good? But here comes what I don't understand. 1 can also be written e^0, so using logarithms we say: e^i2Pi = e^0 -> i2Pi = 0, but this can't be true as i2Pi = 6,28i. I got one possible solution from my math teacher but it did not quite tell me how it could equal 0. Anyone care to give an explenation a shot?
So if you have the e^xi expression and set x = pi, this will equal -1. Adding 2Pi equals 1. So far so good? But here comes what I don't understand. 1 can also be written e^0, so using logarithms we say: e^i2Pi = e^0 -> i2Pi = 0, but this can't be true as i2Pi = 6,28i. I got one possible solution from my math teacher but it did not quite tell me how it could equal 0. Anyone care to give an explenation a shot?
- #5
AUMathTutor
- 498
- 0
My best guess would be that the exponential function is not a bijection over the complex numbers, while it is so over the real numbers. That is, for real numbers,
e^x = e^y <=> x = y is true.
For complex numbers, it must be the case that
e^x = e^y <=> x = y is false.
This shouldn't be so surprising. For the real numbers, you can't say things like...
x^2 = y^2 <=> x = y
sin(x) = sin(y) <=> x = y
...
I suppose it's just an accident that you can't do that for the exponential function over complex numbers.
e^x = e^y <=> x = y is true.
For complex numbers, it must be the case that
e^x = e^y <=> x = y is false.
This shouldn't be so surprising. For the real numbers, you can't say things like...
x^2 = y^2 <=> x = y
sin(x) = sin(y) <=> x = y
...
I suppose it's just an accident that you can't do that for the exponential function over complex numbers.
- #6
dx
Homework Helper
Gold Member
- 2,147
- 50
AUMathTutor said:
You mean injection.
- #7
AUMathTutor
- 498
- 0
Well, injection may be true too, but it's definitely not a bijection either.
I tend to talk in terms of bijection or not bijection. I usually don't delve into things like injectivity and surjectivity. I think that the exponential is neither bijective nor injective over the complex numbers... then again, what do I know?
I tend to talk in terms of bijection or not bijection. I usually don't delve into things like injectivity and surjectivity. I think that the exponential is neither bijective nor injective over the complex numbers... then again, what do I know?
- #8
dx
Homework Helper
Gold Member
- 2,147
- 50
You said exp was a bijection over the reals which it isn't. It's an injection, but not a bijection.
- #9
Count Iblis
- 1,863
- 8
The problem is not with the exponential function, but with the logarithmic function. You need to define a branch cut to define the logarithmic function. This then means that the imaginary part of the Log of a complex number (the so called argument) is a unique number in some interval of length 2 pi.
- #10
belliott4488
- 662
- 1
Yeah, what Count Ibis said:
(At least, I think that's a correct statement; someone please correct me if it's not.)
For the less mathematically versed, this is a little like taking the square root of a number and having to remember that there are actually two roots (+/-). When you take the log of a complex number, there are (potentially) multiple correct results, so you have to specify which one you're choosing.Count Iblis said:
(At least, I think that's a correct statement; someone please correct me if it's not.)
- #11
AUMathTutor
- 498
- 0
"You said exp was a bijection over the reals which it isn't. It's an injection, but not a bijection."
Oops. You're right. I guess that's the price you pay when you get too used to using the general terminology too often.
Oops. You're right. I guess that's the price you pay when you get too used to using the general terminology too often.
- #12
Tibarn
- 27
- 1
The trouble is that the complex exponential function is periodic, so saying that e^x = e^y iff x=y for complex x and y is akin to saying that cos x = cos y iff x=y.
FAQ: Confirming the Euler's Formula: e^{i(a+bx)}=
What is Euler's Formula?
Euler's Formula is a mathematical equation that links complex numbers, trigonometric functions, and the exponential function. It is written as eix = cos(x) + i sin(x), where e is the base of the natural logarithm, i is the imaginary unit, and x is the angle in radians.
What does the Euler's Formula represent?
The Euler's Formula represents the relationship between the exponential function and the trigonometric functions. It shows that a complex number can be expressed in terms of its real and imaginary parts using the exponential function.
How is Euler's Formula confirmed?
Euler's Formula can be confirmed using a method called proof by induction. This involves showing that the formula is true for a specific case (usually n=1) and then proving that if it is true for n=k, it is also true for n=k+1. By repeating this process, we can confirm that the formula holds for all values of n.
Why is Euler's Formula important?
Euler's Formula is important because it has many applications in mathematics, physics, and engineering. It is used to simplify complex calculations involving trigonometric functions and to solve problems in fields such as signal processing, electrical engineering, and quantum mechanics.
Can Euler's Formula be extended to other functions?
Yes, Euler's Formula can be extended to other functions besides eix. For example, the formula can be written as eax = cos(ax) + i sin(ax), where a is a constant. This allows us to represent more complex functions in terms of simpler exponential functions.