In this paper, we will extend Knuth’s up arrow notation \(b \uparrow \uparrow n\) , which for integer n is \(b^{b^{\cdot ^{\cdot ^{b}}}}\) with n b’s, to allow n to be a fraction, or even complex. We do this by first considering the unique complex tetration \(\kappa _b(z)\) , which satisfies the equations \(\kappa _b(z+1) = b^{\kappa _b(z)}\) with \(\kappa _b(0)=1\) , and then fix the value of z as we vary the base b. In particular, we will consider the “half-iterate,” which is the case with \(z=1/2\) , and analytically continue the function into the complex b-plane. To consider “third-iterates” and “fourth-iterates,” we also use 11 other values of z, namely 1/12, 1/6, 1/4, 1/3, 5/12, 7/12, 2/3, 3/4, 5/6, 11/12, and finally \(z=i\) . We make several discoveries by analyzing these functions. First of all, we discover that there is a branch cut at \(b=e^{1/e}\) , and find that as we approach this point, the functions approach one of the known super-exponentials for the base \(e^{1/e}\) . Also, we find that \(b=1\) is also a branch point, and discover what happens as the base approaches 1.
