From ab3e61765a334e43a9528a0873951f2343d5609c Mon Sep 17 00:00:00 2001 From: yum Date: Thu, 25 Sep 2025 01:06:12 +0000 Subject: add stuff --- index.md | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) (limited to 'index.md') diff --git a/index.md b/index.md index 9f319a5..bef51bb 100644 --- a/index.md +++ b/index.md @@ -100,7 +100,9 @@ derivations of each derivative: \frac{\partial \omega}{\partial k} &= \frac{1}{2} [gk \tanh kh]^{-\frac{1}{2}} [g \tanh (kh) + gkh(1 - \tanh ^2 kh] \\ &= \frac{g(\tanh kh + kh(1 - \tanh ^2 kh))}{2 \sqrt{gk \tanh kh}} \\ &= \frac{g \tanh kh + gkh (1 - \tanh^2 kh)}{2 \sqrt{gk \tanh kh}} \\ - &= \frac{1}{2} [\sqrt{g \tanh kh} + \frac {gkh(1 - \tanh^2 kh)}{\sqrt{gk \tanh kh}}] + &= \frac{1}{2} [\sqrt{g \tanh kh} + \frac {gkh(1 - \tanh^2 kh)}{\sqrt{gk \tanh kh}}] \\ + &= \frac {g \tanh kh + gkh(1 - \tanh^2 kh)}{2\sqrt{gk \tanh kh}} \\ + &= \frac {g (\tanh kh + kh \operatorname{sech}^2 kh)}{2\sqrt{gk \tanh kh}} \end{align*} $$ @@ -146,7 +148,8 @@ derivations of each derivative: &= \frac{\partial}{\partial k} (\tanh kh)^{\frac{1}{2}} \\ &= \frac{1}{2} (\tanh kh)^{-\frac{1}{2}} \frac{\partial}{\partial k} \tanh kh \\ &= \frac{1}{2} (\tanh kh)^{-\frac{1}{2}} h(1 - \tanh^2 kh) \\ - &= h \frac{1 - \tanh^2 kh}{2 \sqrt{\tanh kh}} + &= h \frac{1 - \tanh^2 kh}{2 \sqrt{\tanh kh}} \\ + &= h \frac{\operatorname{sech}^2 kh}{2 \sqrt{\tanh kh}} \end{align*} $$ -- cgit v1.2.3