Morera proves holomorphy of $z\mapsto {\int }_0^1{t}^{z}dt$ on $\mathrm{Re}z>1$

Given: The half-plane $\mathrm{\Omega }=\{z:\mathrm{Re}z>1\}$ and the endpoint convention in the example.

For a nonzero complex base $t$ and exponent $z$, the principal power is $t^z=\exp (z\mathrm{Log}t)$; for positive real $t$, $\mathrm{Log}t=\log t$ (Complex logarithms, the principal logarithm, and principal and multivalued complex powers, The natural logarithm as the inverse of the exponential function).

The complex exponential is entire and has derivative equal to itself (The complex exponential is entire and its complex derivative is itself).

For real $x,y$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$ ($\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$).

The complex exponential agrees with the real exponential on the real axis ($\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential).

The real exponential is strictly increasing (The exponential function is strictly increasing).

The natural logarithm is continuous and strictly increasing on the positive reals, with $\log 1=0$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

A complex differentiable function is continuous (Complex differentiability at a point implies continuity there).

The composite of complex differentiable functions is complex differentiable (The chain rule for complex derivatives).

Complex modulus is multiplicative and satisfies the triangle inequality (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

A jointly continuous finite-interval integral of holomorphic parameter slices is holomorphic (A jointly continuous finite-interval parameter integral of holomorphic functions is holomorphic).