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Entire Functions Classification

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StarStarStarStar

f(z) = sin(z)

StarStarStarStar

Order 1, Type 1

StarStarStarStar

f(z) = e^{az} (a \\in \\mathbb{C}, a \\neq 0)

StarStarStarStar

Order 1, Type 1

StarStarStarStar

f(z) = \\sum_{n=0}^{\\infty} z^{2^n}

StarStarStarStar

Order \\infty, Type 2

StarStarStarStar

f(z) = cos(z)

StarStarStarStar

Order 1, Type 1

StarStarStarStar

f(z) = sinh(z)

StarStarStarStar

Order 1, Type 1

StarStarStarStar

f(z) = \\sin(z) + \\sinh(z)

StarStarStarStar

Order 1, Type 1

StarStarStarStar

f(z) = e^{z^2}

StarStarStarStar

Order 2, Type 2

StarStarStarStar

f(z) = \\exp(z + z^{-1})

StarStarStarStar

Order 1, Type 2

StarStarStarStar

f(z) = z^n (n > 0)

StarStarStarStar

Order n, Type 1

StarStarStarStar

f(z) = \\exp(-z^n) (n>1)

StarStarStarStar

Order frac{1}{n}, Type 1

StarStarStarStar

f(z) = \\exp(-e^z)

StarStarStarStar

Order \\infty, Type 3

StarStarStarStar

f(z) = \\sin(\\sqrt{z})

StarStarStarStar

Order frac{1}{2}, Type 1

StarStarStarStar

f(z) = \\Gamma(z)

StarStarStarStar

Infinite Order

StarStarStarStar

f(z) = \mathrm{Ai}(z)

StarStarStarStar

Order frac{3}{2}, Type 2

StarStarStarStar

f(z) = e^z

StarStarStarStar

Order 1, Type 1

StarStarStarStar

f(z) = e^{z^a} (a > 1, a otin \\mathbb{Z})

StarStarStarStar

Order frac{1}{a}, Type 2

StarStarStarStar

f(z) = e^{e^z}

StarStarStarStar

Infinite Order

StarStarStarStar

f(z) = \\log(z)

StarStarStarStar

Undefined

StarStarStarStar

f(z) = \\cos(z^2)

StarStarStarStar

Order 1, Type 2

StarStarStarStar

f(z) = \\cos^{-1}(z)

StarStarStarStar

Undefined

StarStarStarStar

f(z) = \mathrm{Bi}(z)

StarStarStarStar

Order frac{3}{2}, Type 2

StarStarStarStar

f(z) = e^{-z^2}

StarStarStarStar

Order frac{1}{2}, Type 2

StarStarStarStar

f(z) = \\sum_{n=0}^{\\infty} \\frac{z^n}{n!}

StarStarStarStar

Order 1, Type 1

StarStarStarStar

f(z) = \\sum_{n=0}^{\\infty} \\frac{z^n}{n^n}

StarStarStarStar

Order 0, Type 2

StarStarStarStar

f(z) = \\zeta(z)

StarStarStarStar

Undefined

StarStarStarStar

f(z) = z!

StarStarStarStar

Infinite Order

StarStarStarStar

f(z) = z^{1/n} (n>0)

StarStarStarStar

Order frac{1}{n}, Type 1

StarStarStarStar

f(z) = \\tan(z)

StarStarStarStar

Order 1, Type 1

StarStarStarStar

f(z) = \\exp(\\cos(z))

StarStarStarStar

Order 1, Type 1

StarStarStarStar

f(z) = (1 + z/n)^n

StarStarStarStar

Order 1, Type 1

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