Complex Numbers

Exponential functions of a complex variable

The exponential function of a complex variable $z = x + iy$, defined via Euler's formula as $e^z = e^x(\cos y + i \sin y)$, is entirely periodic with period $2\pi i$ and never vanishes. This topic covers its mapping properties, periodicity, and branch relationships.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Euler's Formula and Complex Exponential Mapping Theorem

For any complex number $z = x + iy$ where $x, y \in \mathbb{R}$, the complex exponential function is defined by the infinite series $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$, yielding the closed form $$e^z = e^{x+iy} = e^x (\cos y + i \sin y)$$. Furthermore, $e^z$ is an entire function satisfying $e^{z_1 + z_2} = e^{z_1} e^{z_2}$ and is periodic with fundamental imaginary period $2\pi i$.

📐 Geometric Interpretation

The exponential map takes horizontal lines ($y = c$) in the $z$-plane to rays emanating from the origin in the $w$-plane at angle $c$, and vertical lines ($x = c$) to circles centered at the origin of radius $e^c$. It maps horizontal strips of width $2\pi$ conformally onto the entire punctured complex plane $\mathbb{C} \setminus \{0\}$.

⚙️ Algebraic & Structural Meaning

It establishes a fundamental bridge between transcendental algebraic structures, hyperbolic functions, and trigonometric functions via relations like $\cos z = \frac{e^{iz} + e^{-iz}}{2}$ and $\sin z = \frac{e^{iz} - e^{-iz}}{2i}$, serving as the foundational building block for complex logarithm and power functions.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Analyticity across the finite complex plane	`$z \in \mathbb{C}$ (Entire domain)`	Strictly Mandatory / Necessary	If restricted to non-differentiable domains, Cauchy-Riemann equations fail, destroying single-valued exponential mapping.
Periodicity strip constraint	`$-\pi < y \le \pi$ (Fundamental Strip)`	Strictly Mandatory for Injectivity	Failing to restrict the imaginary part $y$ leads to many-to-one mapping failures and multi-valued ambiguities when inverting to logarithms.
Non-vanishing condition	$z \in \mathbb{C}$	Essential for algebraic closure properties	Attempting to solve $e^z = 0$ yields no solution in $\mathbb{C}$, confirming that the range excludes the origin.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Modulus and Argument Extraction

$$|e^{z}| = e^{\text{Re}(z)}, \quad \arg(e^{z}) = \text{Im}(z) + 2k\pi$$

Rule: Instantly read off the magnitude as $e^x$ and argument as $y$ without converting to polar form manually.

Application: Directly tested in UP PGT to find modulus or real/imaginary parts of complex exponents.

Purely Imaginary Exponent (Euler's Identity)

$$e^{i\theta} = \cos \theta + i \sin \theta, \quad |e^{i\theta}| = 1$$

Rule: Any complex exponential with a purely imaginary exponent lies strictly on the unit circle in the complex plane.

Application: Frequently used in TGT/PGT vector and locus questions involving rotations.

Periodicity Reduction

$$e^{z + 2k\pi i} = e^z, \quad k \in \mathbb{Z}$$

Rule: Strip away integer multiples of $2\pi i$ from the imaginary part before evaluating trigonometric or exponential values.

Application: Saves critical calculation time in complex value simplification problems.

Complex Conjugate of Exponential

$$\overline{e^z} = e^{\bar{z}}$$indices

Rule: The complex conjugate of $e^z$ is simply obtained by conjugating the exponent $z$.

Real and Imaginary Parts Separation

$$\text{Re}(e^{x+iy}) = e^x \cos y, \quad \text{Im}(e^{x+iy}) = e^x \sin y$$

Rule: Instantly decompose complex exponential expressions into Cartesian form for equation solving.

Application: Core technique for solving equations of the form $e^z = a + ib$.

Logarithmic Inverse Bridge

$$\log(z) = \ln|z| + i(\arg z + 2k\pi)$$

Rule: Use the inverse relation to convert exponential equation roots into standard logarithmic forms instantly.

Application: High frequency in UP PGT complex analysis sections.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Assuming $e^z = 1$ implies $z = 0$

Common Pitfall: Candidates forget the periodic nature of the complex exponential and select only $z = 0$.

Counterexample: $e^{2\pi i} = \cos(2\pi) + i \sin(2\pi) = 1$

Why it fails: The equation $e^z = 1$ has infinitely many solutions given by $z = 2k\pi i$ for $k \in \mathbb{Z}$.

🛡️ Exam Defense: Always include the general period term $2k\pi i$ when solving complex exponential equations.

⚠️ COMMON ERROR #02

Zero Value Assumption

Common Pitfall: Assuming that there exists some complex number $z$ such that $e^z = 0$.

Counterexample: $e^x \cos y = 0 \text{ and } e^x \sin y = 0$

Why it fails: Since $e^x \neq 0$ for any real $x$, $\cos y$ and $\sin y$ cannot vanish simultaneously for the same real $y$.

🛡️ Exam Defense: Remember that the range of $e^z$ is $\mathbb{C} \setminus \{0\}$. It never equals zero.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Real Exponential ($e^x$)	$x \in \mathbb{R}$	$e^x > 0$, strictly increasing	Maps real line to positive real axis	Real calculus problems
Complex Exponential ($e^z$)	$z = x + iy \in \mathbb{C}$	$e^x(\cos y + i \sin y)$	Maps strips of width $2\pi$ to punctured plane	Involves $i$ in exponent
Periodicity	$z \to z + 2\pi i$	$e^{z+2\pi i} = e^z$	Repeating cylindrical mapping	Multi-valued inverse questions
Hyperbolic Link	$z = ix$	$e^{ix} = \cos x + i \sin x$	Rotation on unit circle	Trigonometric conversion in $\mathbb{C}$

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If $e^z = -1$, find all possible values of $z$.

(A) $(2k+1)\pi i$(B) $2k\pi i$(C) $\frac{\pi}{2} + 2k\pi i$(D) $(4k+1)\pi i$

📚 Method A: Formal Textbook Derivation

Step 1: Write $e^x \cos y + i e^x \sin y = -1 + 0i$. Step 2: Equate real and imaginary parts: $e^x \cos y = -1$ and $e^x \sin y = 0$. Step 3: From $\sin y = 0$, $y = k\pi$. Step 4: If $y$ is an even multiple of $\pi$, $\cos y = 1 \implies e^x = -1$ (impossible). If $y$ is an odd multiple of $\pi$, $\cos y = -1 \implies e^x = 1 \implies x = 0$. Thus $z = (2k+1)\pi i$.

⚡ Method B: 25-Second Exam Shortcut

Recognize $-1 = e^{i\pi}$. By periodicity, $e^z = e^{i\pi + 2k\pi i} = e^{(2k+1)\pi i}$, giving $z = (2k+1)\pi i$

💡 Tip: Quickly check standard angles where sine is zero and cosine is negative one.

CANONICAL PROBLEM 02 Dual-Method Breakdown

What is the modulus of the complex number $e^{2+3i}$?

(A) $e^3$(B) $e^2$(C) $1$(D) $e^5$

📚 Method A: Formal Textbook Derivation

Step 1: Express $e^{2+3i}$ as $e^2 \cdot e^{3i}$. Step 2: Take modulus: $|e^{2+3i}| = |e^2| \cdot |e^{3i}|$. Step 3: Since $e^2$ is real and positive, $|e^2| = e^2$. Step 4: Since $|e^{3i}| = |\cos 3 + i \sin 3| = \sqrt{\cos^2 3 + \sin^2 3} = 1$, the total modulus is $e^2$.

⚡ Method B: 25-Second Exam Shortcut

Modulus of $e^{x+iy}$ is always $e^{\text{Re}(z)}$. Here $\text{Re}(z) = 2$, so answer is $e^2$.

💡 Tip: Do not waste time expanding trigonometric terms when only the modulus is requested.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: The value of $e^{i\pi/2}$ is equal to:

A 1

B -1

C i

D -i

Derivation & Analysis:

Using Euler's formula, $e^{i\pi/2} = \cos(\pi/2) + i \sin(\pi/2) = 0 + i(1) = i$.

MCQ 2: The fundamental period of the complex exponential function $e^z$ is:

A $2\pi$

B $\pi$

C $2\pi i$

D $\pi i$

Derivation & Analysis:

Since $e^{z + T} = e^z$ requires $e^{T} = 1$, the imaginary period is $T = 2\pi i$.

MCQ 3: The real part of $e^{x+iy}$ is given by:

A $e^x \cos y$

B $e^x \sin y$

C $e^y \cos x$

D $e^y \sin x$

Derivation & Analysis:

By definition, $e^{x+iy} = e^x(\cos y + i \sin y)$, making the real part $e^x \cos y$.

MCQ 4: If $e^z = 1 + i$, then the real part of $z$ is:

A $\ln 2$

B $\frac{1}{2} \ln 2$

C $1$

D $0$

Derivation & Analysis:

$|e^z| = |1+i| = \sqrt{2}$. Since $|e^z| = e^x$, we have $e^x = \sqrt{2} = 2^{1/2}$, so $x = \frac{1}{2}\ln 2$.

MCQ 5: Which of the following statements is FALSE regarding $f(z) = e^z$?

A It is an entire function

B It is periodic with period $2\pi i$

C It takes the value zero at $z = \infty$

D It maps lines parallel to the real axis to rays from the origin

Derivation & Analysis:

$e^z$ has an essential singularity at $z = \infty$ and never equals zero anywhere in the finite complex plane.

MCQ 6: The imaginary part of $e^{2-i}$ is:

A $e^2 \cos 1$

B $-e^2 \sin 1$

C $e^2 \sin 1$

D $-e^2 \cos 1$

Derivation & Analysis:

$e^{2-i} = e^2(\cos(-1) + i \sin(-1)) = e^2(\cos 1 - i \sin 1)$. The imaginary part is $-e^2 \sin 1$.

MCQ 7: The solutions to the equation $e^z + 1 = 0$ are:

A $z = 2k\pi i$

B $z = (2k+1)\pi i$

C $z = k\pi i$

D $z = \frac{(2k+1)\pi}{2} i$

Derivation & Analysis:

$e^z = -1 = e^{i\pi + 2k\pi i}$, which gives $z = (2k+1)\pi i$.

MCQ 8: What is the conjugate of $e^{x+iy}$?

A $e^{x-iy}$

B $-e^{x+iy}$

C $e^{-x-iy}$

D $e^{-x+iy}$

Derivation & Analysis:

$\overline{e^{x+iy}} = \overline{e^x} \cdot \overline{e^{iy}} = e^x (\cos(-y) + i \sin(-y)) = e^{x-iy} = e^{\bar{z}}$.

MCQ 9: The value of $|e^{i\theta}|$ for any real $\theta$ is:

A $1$

B $e^\theta$

C $e^{-\theta}$

D $0$

Derivation & Analysis:

$|e^{i\theta}| = |\cos \theta + i \sin \theta| = \sqrt{\cos^2 \theta + \sin^2 \theta} = 1$.

Q10

MCQ 10: If $e^z = i$, then one particular value of $z$ is:

A $\pi i$

B $\frac{\pi}{2} i$

C $2\pi i$

D $i$

Derivation & Analysis:

$i = e^{i\pi/2}$, hence $z = \frac{\pi}{2}i$ is a primary solution.

Common questions

In the real domain, $e^x = e^{x+T}$ implies $T = 0$ because $\mathbb{R}$ lacks an imaginary component. In the complex domain, Euler's formula introduces trigonometric terms $(\cos y + i \sin y)$ which are periodic with period $2\pi$, imparting a fundamental imaginary period $2\pi i$ to $e^z$.

It can never be zero since $e^x \neq 0$ for any real $x$. However, it can be negative real numbers; for instance, $e^{i\pi} = -1$ and $e^{(2k+1)\pi i} = -1$, showing that negative values occur when the imaginary part is an odd multiple of $\pi$.

Immediately separate real and imaginary parts using $e^{x+iy} = e^x e^{iy}$, apply Euler's identity for the imaginary part, and utilize the periodic property $2\pi i$ to reduce large arguments.

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