```python
import sympy

from sympy import symbols, init_session, sqrt
init_session(quiet=True)
```

    IPython console for SymPy 0.7.6.1 (Python 3.5.4-64-bit) (ground types: python)



```python
tau0, p0, b, P, a, Gamma, delta, sigma, sigma_tau_0 = symbols('tau0 p0 b P a Gamma delta sigma, sigma_tau_0', 
                                                              positive=True, real=True) 
```


```python
T = 2 * tau0 * sqrt(1 - b**2)
tau = 2 * tau0 * p0 / sqrt(1 - b**2)
theta = tau/T
Q = sqrt(Gamma * T) * delta/sigma
sigma_tau = T * sqrt(6 * theta) / Q
sigma_T = T * sqrt(2 * theta) / Q
```

$$ p_0 = \frac{\tau T}{4 \tau_0^2}$$ 

$$ \sigma_{p_0}^2 = \left(\frac{\partial p_0}{\partial \tau}\right)^2 \sigma_\tau^2 + \left(\frac{\partial p_0}{\partial \tau}\frac{\partial p_0}{\partial T}\right) \mathrm{cov}(\tau,T)+\left(\frac{\partial p_0}{\partial T}\right)^2 \sigma_T^2 +\left(\frac{\partial p_0}{\partial \tau_0}\right)^2 \sigma_{\tau_0}^2 $$


$$ \sigma_{p_0}^2 = \left(\frac{T  \sigma_\tau}{4  \tau_0^2} \right)^2 + 
               \left(\frac{\tau  T }{16 \tau_0^4} b  \theta^2 T^2\right) + 
               \left(\frac{\tau  \sigma_T}{4 \tau_0^2} \right)^2 + 
               \left(\frac{\tau  T }{2 \tau_0^3}  \sigma_{\tau_0}\right)^2 $$



```python
sigma_p0_sq = ((T * sigma_tau / 4 / tau0**2)**2 + 
               (tau * T / 16 / tau0**4) * b * theta**2 * T**2 + 
               (tau * sigma_T / 4 / tau0**2)**2 + 
               (tau * T / 2 / tau0**3 * sigma_tau_0)**2)
```


```python
sqrt(sigma_p0_sq).simplify()
```


```python
print(sqrt(sigma_p0_sq).simplify())
```

    sqrt(p0)*sqrt(-(-Gamma*b*delta**2*p0**2*tau0**2*(-b**2 + 1)**(3/2) - 4*Gamma*delta**2*p0*sigma_tau_0**2*(-b**2 + 1)**(5/2) + p0**2*sigma**2*tau0*(b**2 - 1) - 3*sigma**2*tau0*(-b**2 + 1)**3)/(-b**2 + 1)**(5/2))/(sqrt(Gamma)*delta*tau0)



```python

```

70ad509c06231c13b726e54605705ea962160df6
ipynb
Jupyter Notebook
radspots/sympy.ipynb
bmorris3/rms
4239e488f6fae9869782ffbac9f39747b58afdda
MIT
2018-02-13T19:51:47.000Z
2018-02-13T19:51:47.000Z
radspots/sympy.ipynb
bmorris3/rms
4239e488f6fae9869782ffbac9f39747b58afdda
MIT
radspots/sympy.ipynb
bmorris3/rms
4239e488f6fae9869782ffbac9f39747b58afdda
MIT
Qwen/Qwen-72B
1. YES
2. YES