We present timing and spectral analysis of emph{Swift}$-$XRT and emph{RXTE}$-$PCA observations of the transient Be/X-ray pulsar SWIFT J0513.4--6547 during its outburst in 2009 and its rebrightening in 2014. From 2009 observations, short term spin-up rate of the source after the peak of the outburst is found to have about half of the value measured at the peak of the outburst by Coe et al. When the source is quiescent between 2009 and 2014, average spin-down rate of the source is measured to be $sim 1.52 times 10^{-12}$ Hz s$^{-1}$ indicating a surface dipole magnetic field of $sim 1.5 times 10^{13}$ Gauss assuming a propeller state. From 2014 observations, short term spin-down rate of the source is measured to be about two orders smaller than this long-term spin-down rate. The orbit of the source is found to be circular which is atypical for transient Be/X-ray binary systems. Hardness ratios of the source correlate with the X-ray luminosity up to $8.4times 10^{36}$ erg s$^{-1}$ in 3-10 keV band, whereas for higher luminosities hardness ratios remain constant. Pulsed fractions are found to be correlated with the source flux. Overall emph{Swift}$-$XRT and emph{RXTE}$-$PCA energy spectrum of the source fit equally well to a model consisting of blackbody and power law, and a model consisting of a power law with high energy cut-off. From the pulse phase resolved spectra and pulse phase resolved hardness ratios obtained using emph{RXTE}$-$PCA, it is shown that spectrum is softer for the phases between the two peaks of the pulse.
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