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If we assume the "meaning" of ea and if b ∈ Q, b = p q, then we know what does it mean to raise a number by the power p and taking the q − th root of that. That is to say, (ea)p = ea... ea = eap, and (eap)1 q is a number which gives eap when multiplied q -times, and we are done.
Using the base $b=e$, the product rule for exponentials is \begin{gather*} e^ae^b = e^{a+b} \end{gather*} for any numbers $a$ and $b$.
5 Answers. Sorted by: 15. There is no 'only if' because it is not true: eA+B =eAeB. does not necessarily imply [A, B] = 0. One can easily find an example of this using matrices. Here's one: A =(0 0 0 2πi), B =(0 0 1 2πi). [A, B] ≠ 0 but eA+B =eAeB = I. Edit: Let me help with the if part, using a differential equation as OP desires. Compute.
- There is no 'only if' because it is not true: \\begin{align} e^{A+B} = e^A e^B \\end{align} does not necessarily imply $[A,B] = 0$. One can easily fi...
- Define the function $f(u)=e^{uA}e^{uB}$. If $A$ and $B$ commute, you can take the derivatives of the $f(u)$ as follows: $$\\frac{df(u)}{du}=\\frac{de...
- I've never heard of trying to find a differential equation to prove this; I've only done by brute forcing the series expansion. That isn't as bad a...
- Remember that you are working with operators. Since $$e^{A}e^B =(1+\\frac{A}{2}+\\frac{AA}{3!}+...)(1+\\frac{B}{2}+\\frac{BB}{3!}+...)$$ and $$e^{A+B...
- The proof (due to Glauber, given in Messiah) is as follows: consider f(x)= exp(xA)+exp(xB) differentiate it ---> df/dx= Aexp(xA)exp(xB)+exp(xA)ex...
- Version 2019.10.03
- I = C = g(t) = e(A+B)te Bte At
- Y0(t) = AY(t),
Corrections and comments are welcome. For each n n complex matrix A, define the exponential of A to be the matrix eA
for all t. After multiplying by eAteBt on both sides we have eAteBt = e(A+B)t.
is a linear system of ordinary differential equations, then the arguments above imply that
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Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
Detailed step by step solution for simplify (e^a)/ (e^b)
x→−3lim x2 + 2x − 3x2 − 9. Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.
![Example 31 (Supplementary NCERT) - Prove [a b c+d] = [a b c] + [a b d]](https://images.search.yahoo.com/search/images?p=E.A.B.&ei=UTF-8&xargs=0&norw=1&age=1d&th=103&tw=103&imgurl=https%3A%2F%2Fd1avenlh0i1xmr.cloudfront.net%2F0ae05fc1-7cb5-4126-aa91-d5450ced9afe%2Fslide11.jpg&rurl=https%3A%2F%2Fwww.teachoo.com%2F7101%2F748%2FExample-31-%28Supplementary-NCERT%29%2Fcategory%2FExamples%2F&size=66KB&name=Example+31+%28Supplementary+NCERT%29+-+Prove+%5Ba+b+c%2Bd%5D+%3D+%5Ba+b+c%5D+%2B+%5Ba+b+d%5D&oid=2&h=945&w=945&turl=https%3A%2F%2Ftse1.mm.bing.net%2Fth%3Fid%3DOIP.4B_U7aFlsGJIObd9GcRirwHaHa%26pid%3DApi%26rs%3D1%26c%3D1%26qlt%3D95%26w%3D103%26h%3D103&tt=Example+31+%28Supplementary+NCERT%29+-+Prove+%5Ba+b+c%2Bd%5D+%3D+%5Ba+b+c%5D+%2B+%5Ba+b+d%5D&sigr=97WGAFIwKKpb&sigit=nC6ZOZsa.PAN&sigi=8fW.90Xm77mz&sign=MhlYUA0ZS.07&sigt=MhlYUA0ZS.07 "Example 31 (Supplementary NCERT) - Prove [a b c+d] = [a b c] + [a b d]")
