Learn Complex Functions: Interactive Problems and Worked Examples

Complex numbers, conjugates, modulus, and fast algebra

Key idea

A complex number is \(z=x+iy\), where \(x=\Re(z)\) and \(y=\Im(z)\). The complex conjugate is \[ \overline{z}=x-iy. \] The modulus (absolute value) is \[ |z|=\sqrt{x^2+y^2}. \] A key identity is \[ z\overline{z}=|z|^2. \] This is why multiplying by the conjugate helps simplify fractions like \(\frac{1}{a+bi}\).

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Summary

• Conjugate: \(\overline{x+iy}=x-iy\).
• Modulus: \(|x+iy|=\sqrt{x^2+y^2}\).
• Identity: \(z\overline{z}=|z|^2\).

Euler’s formula, \(e^z\), and images of curves

Key idea

Euler’s formula connects exponentials and trigonometry: \[ e^{i\theta}=\cos\theta+i\sin\theta. \] For a general complex number \(z=x+iy\), \[ e^z=e^{x+iy}=e^x e^{iy}=e^x(\cos y+i\sin y). \] This shows two important mapping effects:

• Changing \(x\) scales the magnitude: \(|e^z|=e^x\).
• Changing \(y\) rotates the argument by \(y\) (mod \(2\pi\)).

Also, the exponential is periodic in the imaginary direction: \[ e^{z+2\pi i}=e^z. \] So \(e^z\) is not injective on \(\mathbb{C}\).

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Summary

• \(e^{x+iy}=e^x(\cos y+i\sin y)\): scale by \(e^x\), rotate by \(y\).
• \(e^z\) is not injective on \(\mathbb{C}\) because of \(2\pi i\)-periodicity.
• Inversion \(w=1/z\) sends \(|z|=1\) to \(|w|=1\).

Analytic functions, Cauchy-Riemann equations, and common non-examples

Key idea

Write a complex function as \[ f(z)=u(x,y)+iv(x,y),\quad z=x+iy. \] If \(u\) and \(v\) have continuous first partial derivatives and \(f\) is complex differentiable, then the Cauchy-Riemann equations must hold: \[ u_x=v_y,\qquad u_y=-v_x. \] Many functions fail to be holomorphic because they depend on \(\overline{z}\) or \(|z|\), which mixes \(x\) and \(y\) in a non-analytic way.

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Summary

• Holomorphic \(\Rightarrow\) Cauchy-Riemann (under mild smoothness).
• \(\overline{z}\) and \(|z|^2\) are not holomorphic anywhere on \(\mathbb{C}\).
• Multi-valued functions (like \(z^z\)) require branch choices; they are not analytic on all of \(\mathbb{C}\).

Entire functions, power series, and geometric series in \(\mathbb{C}\)

Key idea

A function is entire if it is holomorphic on all of \(\mathbb{C}\). Classic examples include polynomials and \(e^z\), \(\sin z\), \(\cos z\). Power series in the complex plane behave like real power series: they have a radius of convergence \(R\), and they converge absolutely for \(|z-a|<R\) and diverge for \(|z-a|>R\).

A key example is the geometric series: \[ \sum_{n=0}^{\infty}(z-a)^n \] which converges exactly when \(|z-a|<1\) and then sums to \(\frac{1}{1-(z-a)}\).

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Summary

• Entire means holomorphic on all of \(\mathbb{C}\).
• \(e^z,\sin z,\cos z\), and polynomials are entire.
• \(\sum_{n=0}^{\infty}(z-1)^n\) converges exactly when \(|z-1|<1\).

Classifying singularities: removable, poles, and essential

Key idea

An isolated singularity at \(z=a\) is a point where \(f\) is not holomorphic at \(a\) but is holomorphic on a punctured neighborhood \(0<|z-a|<r\). Three main types:

• Removable singularity: \(f\) can be redefined at \(a\) to become holomorphic (often happens when a factor cancels).
• Pole of order \(m\): \((z-a)^m f(z)\) is holomorphic and nonzero at \(a\). A simple pole is \(m=1\).
• Essential singularity: neither removable nor a pole; Laurent series has infinitely many negative-power terms.

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Summary

• \(e^{1/z}\) and \(\sin(1/z)\) have essential singularities at \(z=0\).
• \(\frac{\sin z}{z}\) has a removable singularity at \(0\) (define value as \(1\)).
• \(\frac{\sin(1/z)}{z}\) has an essential singularity at \(0\).

Residues at simple poles and fast computations

Key idea

If \(f\) has a simple pole at \(z=a\), then its residue is \[ \operatorname{Res}(f,a)=\lim_{z\to a}(z-a)f(z). \] A rational function \(\frac{p(z)}{q(z)}\) typically has poles where \(q(z)=0\) (if \(p(a)≠ 0\) and the zero is simple).

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Summary

• Simple-pole residue: \(\operatorname{Res}(f,a)=\lim_{z\to a}(z-a)f(z)\).
• \(\tan z\) has simple poles at \(\frac{\pi}{2}+k\pi\).

Contour integrals, angle preservation, and final recap

Core contour integral fact

For the unit circle \(|z|=1\), a fundamental identity is: \[ \oint_{|z|=1} z^n\,dz = \begin{cases} 2\pi i, & n=-1,\\ 0, & n≠ -1, \end{cases} \] where \(n\) is an integer. In particular, \[ \oint_{|z|=1} z^2\,dz = 0. \]

Angle preservation (conformality)

Holomorphic functions with nonzero derivative preserve angles between smooth curves (they are conformal). The map \(f(z)=\overline{z}\) is not holomorphic; it reflects the plane and does not preserve oriented angles, so it is not conformal.

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Final recap

• Conjugate and modulus: \(\overline{x+iy}=x-iy\), \(|x+iy|=\sqrt{x^2+y^2}\), \(z\overline{z}=|z|^2\).
• Exponential: \(e^{x+iy}=e^x(\cos y+i\sin y)\), and \(e^{z+2\pi i}=e^z\) (not injective on \(\mathbb{C}\)).
• Holomorphic: check Cauchy-Riemann; \(\overline{z}\) and \(|z|^2\) are not holomorphic anywhere.
• Entire: holomorphic on all of \(\mathbb{C}\); examples include \(e^z\), \(\sin z\), \(\cos z\), polynomials.
• Singularities: removable (e.g. \(\sin z/z\) at \(0\)), poles, essential (e.g. \(e^{1/z}\) at \(0\)).
• Residue at a simple pole: \(\operatorname{Res}(f,a)=\lim_{z\to a}(z-a)f(z)\).
• Contour integral: \(\oint_{|z|=1} z^2\,dz=0\).