A little help understanding math....
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08-29-2021, 12:20 AM
(This post was last modified: 01-26-2024 03:19 PM by Albert Chan.)
Post: #9
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RE: A little help understanding math....
Prove: if 0 ≤ θ < pi/2, Re(atan(exp(i*θ))) = pi/4
atan(z) = atanh(z*i)/i = ln((1+i*z)/(1-i*z)) / (2*i) Let z = exp(i*θ) = cos(θ) + i*sin(θ) (1+i*z)/(1-i*z) = ((1-sin(θ)) + i*cos(θ)) / ((1+sin(θ)) - i*cos(θ)) = i * cos(θ) / (1 + sin(θ)) // flip sin/cos = i * sin(pi/2-θ) / (1 + cos(pi/2-θ)) // tan(α/2) = sin(α) / (1 + cos(α)) = i * tan(pi/4-θ/2) For 0 ≤ θ < pi/2, imaginery part is positive. Re(atan(exp(i*θ))) = arg(i*tan(pi/4-θ/2)) / 2 = pi/4 With signed zero, we have atan(±0 + i) = ±pi/4 + Inf*i Code: Complex Catan(Complex z) Top branch, atan(t = -1/z) ≈ t - t^3/3 + t^5/5 - ... ≈ t, is because of trig identities. S2k+1 = sin((2k+1)θ)/sin(θ) = 2 cos((2k)θ) + 2 cos((2k-2)θ) + 2 cos((2k-4)θ) + ... + 1 C2k+1 = cos((2k+1)θ)/cos(θ) = 2 cos((2k)θ) - 2 cos((2k-2)θ) + 2 cos((2k-4)θ) − ... + (-1)^k --> |S2k+1| ≤ 2k+1 --> |C2k+1| ≤ 2k+1 t = ε * cis(θ) t^n/n = ε^n/n * cis(nθ) |re(t^3/3)/re(t)| = ε^2 * |C3|/3 ≤ ε^2 |im(t^3/3)/im(t)| = ε^2 * |S3|/3 ≤ ε^2 |re(t^5/5)/re(t)| = ε^4 * |C5|/5 ≤ ε^4 |im(t^5/5)/im(t)| = ε^4 * |S5|/5 ≤ ε^4 ... ε^2 below machine epsilon --> t - t^3/3 + t^5/5 - ... ≈ t |
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Messages In This Thread |
A little help understanding math.... - Thomas Klemm - 12-16-2013, 11:56 PM
RE: A little help understanding math.... - Paul Dale - 12-17-2013, 10:19 AM
RE: A little help understanding math.... - Namir - 12-17-2013, 01:17 PM
RE: A little help understanding math.... - Thomas Klemm - 12-17-2013, 03:04 PM
RE: A little help understanding math.... - Albert Chan - 08-15-2021, 03:48 AM
RE: A little help understanding math.... - Albert Chan - 08-15-2021, 12:25 PM
RE: A little help understanding math.... - Albert Chan - 08-26-2021, 02:31 PM
RE: A little help understanding math.... - Albert Chan - 08-26-2021, 06:16 PM
RE: A little help understanding math.... - Albert Chan - 08-29-2021 12:20 AM
RE: A little help understanding math.... - Albert Chan - 06-14-2023, 04:55 PM
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