- #1
ILoveParticlePhysics
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How do we know that e^πi= -1 if all numbers here are basically undefined/irrational?
How is Euler's Identity proven using differential equations?
- Context: Undergrad
- Thread starter ILoveParticlePhysics
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Discussion Overview
The discussion centers around the proof of Euler's identity, specifically the equation \( e^{\pi i} = -1 \), using differential equations and power series. Participants explore the definitions and properties of the exponential function, particularly in relation to complex numbers and trigonometric functions.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Exploratory
Main Points Raised
- One participant questions the validity of \( e^{\pi i} = -1 \) by suggesting that the numbers involved are undefined or irrational.
- Another participant suggests starting from the power series representation of the exponential function, \( e^{iz} = \sum_{k=0}^{\infty} \frac{i^k z^k}{k!} \), and proposes splitting the sum into odd and even terms to relate it to sine and cosine functions.
- A third participant reiterates the question about the irrationality of the numbers involved in the identity, emphasizing the generalization of the exponential function through power series.
- One participant provides a proof using differential equations, stating that both \( f(x) = e^{ix} \) and \( g(x) = \cos{x} + i\sin{x} \) satisfy the same differential equation and initial condition, leading to the conclusion that they are the same function.
- This participant concludes by suggesting to evaluate \( f(\pi) = g(\pi) \) to derive Euler's identity.
Areas of Agreement / Disagreement
Participants express differing views on the nature of the numbers involved in Euler's identity, with some questioning their definitions while others focus on the mathematical proof. The discussion includes both supportive and critical perspectives, indicating that no consensus has been reached regarding the foundational aspects of the identity.
Contextual Notes
Some limitations include the potential ambiguity in the definitions of the numbers involved and the reliance on the uniqueness of solutions to differential equations, which may not be universally accepted without further clarification.
- #2
etotheipi
Hey, I'm not irrational! 
Anyway, start from$$e^{iz} = \sum_{k=0}^{\infty} \frac{i^k z^k}{k!} $$and split this sum into odd and even terms. See if you can get it in terms of sines and coses.
Anyway, start from$$e^{iz} = \sum_{k=0}^{\infty} \frac{i^k z^k}{k!} $$and split this sum into odd and even terms. See if you can get it in terms of sines and coses.
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- #3
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Generalisations of the exponential function can usually be defined via the power series method above. That's how we can also define the exponential of a matrix.ILoveParticlePhysics said:
- #4
Gaussian97
Homework Helper
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In case you are more familiar with differential equations, there's a proof of the fact that ##e^{ix}=\cos{x}+i\sin{x}## that it's really nice. If we define
$$f(x)=e^{ix}, \qquad g(x)=\cos{x}+i\sin{x}$$
It is really easy to see that both satisfy the differential equation
$$f'(x) = i f(x), \qquad g'(x) = i g(x)$$
Furthermore, both satisfy the initial condition
$$f(0)=g(0)=1$$
Because ##f(0)=e^{0}=1##.
Then, since these two functions satisfy the same differential equation with the same initial condition, by the uniqueness of the solutions (Picard's theorem) they must be the same function. QED
Then just evaluate ##f(\pi)=g(\pi)## to obtain Euler's identity.
$$f(x)=e^{ix}, \qquad g(x)=\cos{x}+i\sin{x}$$
It is really easy to see that both satisfy the differential equation
$$f'(x) = i f(x), \qquad g'(x) = i g(x)$$
Furthermore, both satisfy the initial condition
$$f(0)=g(0)=1$$
Because ##f(0)=e^{0}=1##.
Then, since these two functions satisfy the same differential equation with the same initial condition, by the uniqueness of the solutions (Picard's theorem) they must be the same function. QED
Then just evaluate ##f(\pi)=g(\pi)## to obtain Euler's identity.
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