Power series can be tuned to match both the near and far behavior of a function.

Standard math formulas called power seriespower seriesA way to write a function as an infinite sum of terms like x, x², x³, each multiplied by a constant. approximate curves by adding up powers of a variable, matching how the curve bends at a single starting point. However, these basic series often fail far away from that center point. To fix this, researchers replaced the standard variable with a new adjustable function that contains flexible settings. This change lets the series keep matching derivatives at the starting point while giving researchers knobs to tweak how it behaves farther away. They designed six specific versions of this function, each having five adjustable parameters. Testing showed that this technique can speed up how fast the approximations settle, expand where they work, imitate tricky mathematical features like branch points and branch cutsbranch points and branch cutsSpecial points or curves where a mathematical function becomes multi-valued, like the square root function having two possible values for one input., and yield new formulas for partial Bell polynomialsBell polynomialsSpecial polynomials that appear when computing derivatives of composed functions, used here as a tool in the expansion formulas..

Why it matters

Standard power series are great at approximating functions near a point but often fail far away. This method lets you adjust the expansion so it stays accurate over a wider range, improving convergence and handling tricky features like branch cuts. It could make mathematical modeling more flexible in physics and engineering.

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