Time transfer from proper to coordinate: apparent Special / General Relativity mismatching in theory In SR we've learned that the time dilation for an observer moving clock w.r.t one fixed in a frame at rest is
$$\tau = \gamma \tau_0 = \frac{\tau_0}{\left(1-v^2/c^2\right)^{1/2}}$$
ref: "Special Relativity - A.P. French" and many others
In this case being gamma > 1, it implies delta t < delta tau
No moving to GR, the basic starting expression for calculating the elapsed coordinate time from proper time for a observer clock located in a mass gravitational field and moving with velocity v w.r.t a frame at rest in the body mass center is (approximating square root at first order for v << c)
$$\Delta t = \int_A^B \left(1+\frac 1 {c^2} U + \frac{1}{2c^2} v^2\right)d\tau$$
ref: "Relativistic time transfer - ITU-R TF.2118-0" and many other
To note that all terms in the integral are positive, also excluding the presence of gravity (U=0), meaning that it would always result delta t > delta tau
This is an opposite result w.r.t. SR expression!
Can anyone clarify this (apparent) contradiction? Thanks in advance.
 A: 
Can anyone clarify this (apparent) contradiction?

It's just different nomenclature. There is no contradiction.
French's equation 4-5 is
$$\tau = \gamma \tau_0 = \frac {\tau_0} {\left(1-v^2/c^2\right)^{1/2}}$$
Note that French's equation 4-5 uses two taus, $\tau$ and $\tau_0$, to represent the time difference between two events as measured by two different observers. The latter ($\tau_0$) represent the time difference as measured by an observer at rest with respect to the two events. The former ($\tau$) represents the time difference as measured by an observer moving with respect to the stationary observer.
French's $\tau$ is coordinate time ($\Delta t$ in more modern nomenclature) while his $\tau_0$ is proper time ($\Delta \tau$ in more modern nomenclature). A more modern way to write French's equation 4-5 is thus
$$\Delta t = \gamma \Delta \tau = \frac {\Delta \tau} {\left(1-v^2/c^2\right)^{1/2}}$$
With this modernized rewrite it is obvious that there is no contradiction.
A: Using bionomial approximation, French's equation equals:
$$\tau = \gamma \tau_0 = \frac {\tau_0} {\left(1-v^2/c^2\right)^{1/2}}\approx{\left(1+\frac{v^2}{2c^2}\right)\tau_0}$$
The second equation introduced by you, substituting $U=0$, also implies:
$$\Delta t\approx{\left(1+\frac{v^2}{2c^2}\right)\Delta\tau}$$
There is no contradiction, I think!
@Gianni:

... the elapsed coordinate time from proper time for an observer clock located in a mass gravitational field and moving with velocity $v$ w.r.t a frame at rest in the body mass center ...

However, remember that an observer located at the center of a planet measures that the clock located on the surface of the planet runs faster. I am doubtful that the boldfaced sentence may not indicate the Schwarzschild observer but rather the observer at the center of the planet. This can justify the difference between the two equations if you think $Δτ$ does not match $τ_0$.
