## review derivation: Euler equation proof

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
10 make expr power
1. 4923339482; locally 3784785:
$$i x = \log(y)$$
$$pdg_{1464} pdg_{4621} = \frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}}$$
1. 0006656532:
$$e$$
$$pdg_{2718}$$
1. 9482923849; locally 9587572:
$$\exp(i x) = y$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{1452}$$
LHS diff is -pdg2718**(pdg1464*pdg4621) + exp(pdg1464*pdg4621) RHS diff is pdg1452 - pdg2718**(log(pdg1452)/log(10)) 4923339482:
9482923849: error for dim with 9482923849
4923339482:
9482923849: N/A
7 indefinite integrate RHS over
1. 9482113948; locally 8883737:
$$\frac{dy}{y} = i dx$$
$$pdg_{4621}$$
1. 0004264724:
$$y$$
$$pdg_{1452}$$
1. 9482943948; locally 9984877:
$$\log(y) = i dx$$
$$\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{4621} pdg_{9199}$$
Nothing to split 9482113948:
9482943948:
9482113948:
9482943948:
5 multiply both sides by
1. 9848294829; locally 9038289:
$$\frac{d}{dx} y = y i$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{1452} pdg_{4621}$$
1. 0003954314:
$$dx$$
$$pdg_{9199}$$
1. 9848292229; locally 1111289:
$$dy = y i dx$$
$$pdg_{5842} = pdg_{1452} pdg_{4621} pdg_{9199}$$
LHS diff is -pdg5842 RHS diff is 0 9848294829:
9848292229:
9848294829:
9848292229:
3 factor out X from RHS
1. 9429829482; locally 1838300:
$$\frac{d}{dx} y = -\sin(x) + i\cos(x)$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \cos{\left(pdg_{1464} \right)} - \sin{\left(pdg_{1464} \right)}$$
1. 0007563791:
$$i$$
$$pdg_{4621}$$
1. 9482984922; locally 2948271:
$$\frac{d}{dx} y = (i\sin(x) + \cos(x)) i$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \left(pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\right)$$
LHS diff is 0 RHS diff is -(pdg4621**2 + 1)*sin(pdg1464) 9429829482:
9482984922:
9429829482:
9482984922:
8 indefinite integrate RHS over
1. 9482943948; locally 9984877:
$$\log(y) = i dx$$
$$\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{4621} pdg_{9199}$$
1. 0006563727:
$$x$$
$$pdg_{1464}$$
1. 4928239482; locally 3747585:
$$\log(y) = i x$$
$$\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{1464} pdg_{4621}$$
no check performed 9482943948:
4928239482:
9482943948:
4928239482:
2 differentiate with respect to
1. 9492920340; locally 1029383:
$$y = \cos(x)+i \sin(x)$$
$$pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 0007636749:
$$x$$
$$pdg_{1464}$$
1. 9429829482; locally 1838300:
$$\frac{d}{dx} y = -\sin(x) + i\cos(x)$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \cos{\left(pdg_{1464} \right)} - \sin{\left(pdg_{1464} \right)}$$
no check performed 9492920340: error for dim with 9492920340
9429829482:
9492920340: N/A
9429829482:
6 divide both sides by
1. 9848292229; locally 1111289:
$$dy = y i dx$$
$$pdg_{5842} = pdg_{1452} pdg_{4621} pdg_{9199}$$
1. 0009877781:
$$y$$
$$pdg_{1452}$$
1. 9482113948; locally 8883737:
$$\frac{dy}{y} = i dx$$
$$pdg_{4621}$$
Nothing to split 9848292229:
9482113948:
9848292229:
9482113948:
1 declare initial expr
1. 9492920340; locally 1029383:
$$y = \cos(x)+i \sin(x)$$
$$pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 9492920340: error for dim with 9492920340
9492920340: N/A
4 substitute RHS of expr 1 into expr 2
1. 9492920340; locally 1029383:
$$y = \cos(x)+i \sin(x)$$
$$pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
2. 9482984922; locally 2948271:
$$\frac{d}{dx} y = (i\sin(x) + \cos(x)) i$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \left(pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\right)$$
1. 9848294829; locally 9038289:
$$\frac{d}{dx} y = y i$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{1452} pdg_{4621}$$
valid 9492920340: error for dim with 9492920340
9482984922:
9848294829:
9492920340: N/A
9482984922:
9848294829:
12 declare final expr
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 4938429483: error for dim with 4938429483
4938429483: N/A
9 swap LHS with RHS
1. 4928239482; locally 3747585:
$$\log(y) = i x$$
$$\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{1464} pdg_{4621}$$
1. 4923339482; locally 3784785:
$$i x = \log(y)$$
$$pdg_{1464} pdg_{4621} = \frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}}$$
valid 4928239482:
4923339482:
4928239482:
4923339482:
11 substitute RHS of expr 1 into expr 2
1. 9482923849; locally 9587572:
$$\exp(i x) = y$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{1452}$$
2. 9492920340; locally 1029383:
$$y = \cos(x)+i \sin(x)$$
$$pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
valid 9482923849: error for dim with 9482923849
9492920340: error for dim with 9492920340
4938429483: error for dim with 4938429483
9482923849: N/A
9492920340: N/A
4938429483: N/A
Physics Derivation Graph: Steps for Euler equation proof

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
2718 constant \exp
$$\exp$$
['real']
e 2.71828   unitless
8
1452 variable y
$$y$$
['real']
13
9199 variable dx
$$dx$$
['real']
• length: 1
15
4621 variable i
$$i$$
['imaginary']
imaginary unit
74
5842 variable dy
$$dy$$
['real']
• length: 1
differential displacement along y axis
• str_note
2
1464 variable x
$$x$$
['real']
140
MESSAGE:
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