Let \(k\) be an algebraically closed field of characteristic \(p>2\); let \(W(k)\) be the ring of Witt vectors over \(k\). Let \(B\) be a \(W(k)\)-algebra. \(B\) is said to be \(p\)-adic provided it is closed and separated in the \(p\)-adic topology. Let \(E\) be an elliptic curve over a \(p\)-adic \(B\) such that the formal group \(\hat E\) of \(E\) is isomorphic to \(\hat G_m\times B\); let \(\mathcal L\) be an ample, totally symmetric line bundle on \(E\); let \(s\) be a section of \(\mathcal L\). Then one can associate to \(s\) a theta function \(\Theta_s\) on the formal group \(\hat E\). If \(B=W(k)((q))\) and \(E\) is the Tate curve then \(\Theta\) can be written explicitly; for example, for appropriate \(\mathcal L\) and \(s\), \(\Theta_s(w)=\sum^{n=\infty}_{n=-\infty}q^{n^2}w^n\) where \(w\) is a multiplicative parameter on \(\hat E\). The author treats the problem to exhibit these theta functions explicitly in the case of other elliptic curves; in particular, in the case of the universal lifting of an ordinary elliptic curve. He achieves a representation of \(\Theta_s\) as an integral: If the degree of \(\mathcal L\) is prime to \(p\), there is a finite abelian group \(D\) and a measure on \(D\subset\mathbb{Q}_p\) such that \(\Theta_s(w)=\int_{m\in D\times\mathbb{Q}_p}f(m)w^ m\,d\mu(m)\) where \(f\) is locally constant with compact support. Moreover the author proves that \(\Theta_s\) satisfies a heat equation. By combining the integral formula for theta functions with the heat equation, he obtains the following formula for theta: Let \(E_0\) be an ordinary elliptic curve over \(k\) and let \(E/W(k)[[q]]\) be its universal lifting. Given an isomorphism \(\hat E \simeq G_m\times W(k)[[ q]]\), let \(w\) be a multiplicative parameter on \(\hat E\); let \(q\) be the corresponding Serre-Tate parameter. Let \(\mathcal L\) be a line bundle of degree 4 on \(E\); then, after adjoining the fourth root of unity to \(W(k)\), for appropriate sections \(s\in \Gamma (\mathcal L)\) one has \(\Theta_s = \int_{m\in Z_1}(1+q)^{m^2}w^{-2m}\,d\mu (m)\).