source/reference:
https://www.youtube.com/channel/UCaTLkDn9_1Wy5TRYfVULYUw/playlists

Complex Function

Complex Logarithm Function

Given 𝑧∈ℂ\{0},, find 𝑤∈ℂ such that ℯ^𝑤=𝑧

𝑧=|𝑧|ℯ^𝑖𝜃, then ℯ^𝑤=|𝑧|ℯ^𝑖𝜃

Next, write 𝑤=𝑢+𝑖𝑣. Then ℯ^𝑢ℯ^𝑖𝑣=|𝑧|ℯ^𝑖𝜃

Thus ℯ^𝑢=|𝑧| and ℯ^𝑖𝑣=ℯ^𝑖𝜃, so 𝑢=ln|𝑧| and 𝑣=𝜃+2𝑘𝜋=arg𝑧

Therefore, 𝑤=ln|𝑧|+𝑖arg𝑧

By definition. For 𝑧≠0,

Log𝑧=ln|𝑧|+𝑖Arg 𝑧, the principal branch of logarithm,

and

log𝑧=ln|𝑧|+𝑖arg𝑧, a multi-valued function =Log𝑧+2𝑘𝜋𝑖, 𝑘∈ℤ

Examples

Log𝑧=ln|𝑧|+𝑖Arg𝑧

• Log1=ln|1|+𝑖Arg1=0
• Log𝑖=ln|𝑖|+𝑖Arg𝑖=0+𝑖𝜋2=𝑖𝜋2
• Log(-1)=ln|-1|+𝑖Arg-1=0+𝑖𝜋=𝑖𝜋
• Log(1+𝑖)=ln|1+𝑖|+𝑖Arg(1+𝑖)=ln√2+𝑖𝜋4

Continuity of the Logarithm Function

For the logarithm function, Log𝑧=ln|𝑧|+𝑖Arg𝑧

• z⟼|z| is continuous in ℂ
• z⟼ln|z| is continuous in ℂ\{0}
• z⟼Argz is continuous in ℂ\(−∞,0]
• Thus, Logz is continuous in ℂ\(−∞,0]
• However,
  • as z→−𝑥∈(−∞,0) from above, Logz→ln𝑥+𝑖𝜋, and
  • as z→−𝑥 from below, Logz→ln𝑥−𝑖𝜋,
  so Logz is not continuous on (−∞,0) (and not defined at 0.

Derivative of Logarithm Function

By Fact. The principal branch of logarithm, Logz, is analytic in ℂ\(−∞,0].

The derivative:

ℯLogz=z 𝑑𝑑z(ℯLogz)=𝑑𝑑zz ℯLogz⋅𝑑𝑑zLogz=1 𝑑𝑑zLogz=1ℯLogz=1z

More General Theorem

By theorem. Suppose that 𝑓:𝑈→ℂ is an analytic function and there exists a continuous function 𝑔:𝐷→𝑈 from some domain 𝐷⊂ℂ into 𝑈 such that 𝑓(𝑔(z))=z for all z∈𝐷. Then 𝑔 is analytic in 𝐷, and

𝑔′(z)=1𝑓′(𝑔(z)) for z∈𝐷

Application 1

Let 𝑓:ℂ→ℂ, 𝑓(z)=z². Then 𝑓′(z)=2z.

Let 𝑔:ℂ\(−∞,0]→ℂ, 𝑔(z)=√z be the principal branch of the square root (=√|z|⋅ℯ^𝑖Argz2). Then

• 𝑓(𝑔(z))=z for all z∈𝐷=ℂ\(−∞,0]
• 𝑔 is continuous in 𝐷, thus

• 𝑔 is analytic in 𝐷, and

  𝑔′(z)=1𝑓′(𝑔(z))  =12𝑔(z)  =12√2

Application 2

Let 𝑓:ℂ→ℂ, 𝑓(z)=z². Then 𝑓′(z)=2z.

Let ℎ:ℂ\[0,∞)→ℂ, ℎ(z)={√z, imz≥0−√z, imz<0 (=√|z|⋅ℯ^{𝑖Argz2(+𝑖𝜋)}=±√|z|⋅ℯ^𝑖Argz2 =±√|z|). Then

• 𝑓(ℎ(z))=z for all z∈𝐷=ℂ\[0,∞)
• ℎ is continuous in 𝐷, thus

• ℎ is analytic in 𝐷, and

  ℎ′(z)=1𝑓′(ℎ(z))  =12ℎ(z)

Terminology

Let 𝑓:𝑈→𝑉 be a function.

• 𝑓 is injective, also called 1-1, provided that 𝑓(𝑎)≠𝑓(𝑏) whenever 𝑎, 𝑏∈𝑈 with 𝑎≠𝑏.
• 𝑓 in surjective, also called onto, provided that for every 𝑦∈𝑉 there exists and 𝑥∈𝑈 such that 𝑓(𝑥)=𝑦.
• 𝑓 is a bijection, also called 1-1 and onto, if 𝑓 is both injective and surjective.

Examples:

• 𝑓:{z∈ℂ|Rez>0}→ℂ\(−∞,0], 𝑓(z)=z² is a bijection
• 𝑓:ℂ→ℂ, 𝑓(z)=z² is not injective but is surjective.
• 𝑓:ℂ\(−∞,0]→ℂ, 𝑓(z)=√z is injective but not surjective.